828  Broadway 


aiitorma 
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REESE   LIBRARY 


UNIVERSITY    OF    CALIFORNIA. 


UMH/FPSITV  OF  HA? 


A   TREATISE 


BELTS  AND  PULLEYS. 


EMBRACING 


FULL  EXPLANATIONS  OF  FUNDAMENTAL  PRINCIPLES;    PROPER 

DISPOSITION  OF  PULLEYS;  RULES,  FORMULAS,  AND  TABLES 

FOR    DETERMINING    WIDTHS    OF    LEATHER    AND    VUL- 

CANIZED-RUBBER    BELTS    AND    BELTS    RUNNING 

OVER  COVERED  PULLEYS;    STRENGTH    AND 

PROPORTIONS  OF  PULLEYS,  DRUMS,  ETC. 

TOGETHER  WITH 

THE  PRINCIPLES  OF  AND  NECESSARY  RULES  FOR  ROPE-GEARING 

AND  TRANSMISSION   OF  POWER  BY  MEANS  OF 

METALLIC  CABLES. 


BY 

J.  HOWARD    CROMWELL,  Pn.B., 

AUTHOR  OF  A  TREATISE  ON  TOOTHED  GEARING. 


NEW   YORK: 
JOHN   WILEY   AND    SONS. 

1888, 


Copyright,  1885, 
BY  TQHN   WILEY   &   SONS. 


PREFACE. 


IN  the  manufacture  of  modern  machinery,  which  in  the 
great  majority  of  cases  embodies  a  vast  deal  of  careful  study 
and  precise  calculation,  there  is  probably  no  one  element  which 
enters  as  largely  into  the  calculations  and  forms  as  important 
a  part  in  the  daily  operations  in  the  machine-shop  as  the  end- 
less belt  for  the  transmission  of  power.  The  lathe,  the  drill, 
the  planer,  the  shaping-machine — in  short,  almost  without  ex- 
ception, all  machine-tools — are  commonly  driven  from  the 
shop-shaft  by  means  of  belts  and  pulleys;  and  we  can  scarcely 
glance  into  a  shop  or  factory  of  any  description  without  en- 
countering a  mass  of  belts  which  seem  at  first  sight  to  mo- 
nopolize every  nook  in  the  building  and  leave  little  or  no  room 
for  anything  else. 

Notwithstanding  the  countless  thousands  of  belts  for  trans- 
mission in  use  and  constantly  being  replaced  in  the  shops  and 
factories  of  America;  notwithstanding  the  fact  that  many 
thousands  of  dollars  are  consumed  every  year  by  the  rapid 
wear  and  destruction  of  our  machine-belts,  and  the  immense 
field  thereby  opened  for  the  practical  study  and  application  of 
the  principles  of  economy  in  this  connection — there  is  no 
branch  of  machine-construction  which  is  to-day  in  as  crude 
and  unsatisfactory  a  state  of  development  as  this  all-important 
transmission  by  belt  and  pulley.  Strange  as  it  may  seem,  it  is 


IV  PREFACE. 

nevertheless  true,  that  there  is  scarcely  a  machine-shop  in 
America  which  can  definitely  and  correctly  calculate  the  proper 
width  of  a  leather  belt  which  will  safely  transmit  a  given  horse- 
power. Nor  are  the  manufactures  of  machine-belting  in  any 
degree  in  advance  of  the  shops,  for  I  have  never  yet  seen  the 
manufacturer  who  has  any  better  solution  for  this  apparently 
simple  problem  than  his  own  "judgment." 

Having  taken  the  pains  to  write  to  a  large  number  of  the 
best-known  machine-shops  and  belt-manufacturers  throughout 
the  country,  asking  for  information  concerning  belting,  and  in 
every  case  having  received  an  answer  to  the  communication,  I 
am  compelled  to  assert  that  among  all  the  letters  received  not 
a  single  one  contained  any  definite  information  on  the  subject. 
As  specimen  answers  to  these  letters  I  may  quote  the  following 
extracts : 

"  We  have  no  particular  method  of  calculating  widths  of 
belts  aside  from  tables  found  in  books  of  reference."  "There 
is  no  rule  for  the  width  of  belting  that  we  know  of :  it  is  always 
determined  by  the  width  of  the  pulley  upon  which  it  is  to  run." 
"  We  determine  the  width  of  belts  more  by  experience  than  by 
any  fixed  rule."  "  We  always  try  and  make  the  strain  as  light, 
in  pounds  per  inch  of  width,  as  possible,  and  when  we  are 
limited  for  room  we  use  double  belts.  100  pounds  per  inch 
of  width  is  ^about  the  ultimate  strength  of  transmission,  and  if 
you  can  reduce  the  working  strain  to  50  pounds,  it  means  long 
life  to  the  belt."  "  It  is  difficult  to  give  any  positive  rule  about 
belting  that  would  apply  to  all  cases."  From  one  of  the  largest 
and  best-known  belt- manufacturing  concerns  in  the  country 
comes  the  following :  "  We  have  no  rules  or  formulas  for  esti- 
mating the  power  of  belts  other  than  those  given  in  works  on 
mechanical  engineering,  nor  do  we  apply  these  strictly.  It  is 


PREFA  CE.  V 

very  much  a  question  of  judgment.  .  .  .  You  will  consider  this 
letter  very  indefinite,  but  we  do  not  know  how  to  make  it  less 
so."  Here  are  extracts  from  a  letter  received  from  another 
well-known  belt-maker:  "We  wish  to  express  the  fear  that 
what  we  have  to  say  will  be  disappointing  to  you,  to  say  the 
least.  ...  As  to  the  horse-power,  we  have  no  rule.  .  .  .  We 
have  made  no  tests  of  the  tensile  strength  of  leather,  for  the 
reason  that  we  do  not  consider  it  a  matter  of  any  importance. 
.  .  .  We  have  made  no  efforts  to  obtain  the  coefficient  of  fric- 
tion. .  .  .  When  we  can  obtain  a  homogeneous  material  which 
will  be  easily  workable  and  a  perfect  substitute  for  leather,  the 
manufacture,  sale,  use,  and  study  of  belting  may  begin  to  be  a 
matter  of  satisfaction  ;  in  the  meanwhile  they  are  puzzling,  if 
not  indeed  exasperating." 

These  extracts  (many  more  of  similar  nature  might  be  given) 
show  almost  no  knowledge  at  all,  on  the  part  of  our  great  belt- 
manufacturers  and  machine-shops,  concerning  the  subject;  and 
worse  still  in  some  cases,  that  little  or  no  effort  has  been  made 
to  obtain  any  knowledge  other  than  that  of  rough  guesswork 
and  rule  of  thumb.  Small  wonder  is  it,  then,  that  the  ordinary 
mechanic's  practical  knowledge  of  the  subject  is  infinitely  small. 
Several  of  the  parties  above  referred  to  state  that  they  use  the 
rules  found  in  the  various  books  of  reference ;  let  us  look  over 
some  of  these  works  and  endeavor  to  reach  fair  conclusions 
concerning  the  rules  and  formulas  in  common  use  to-day. 

Arnold,  in  his  "  Mechanical  Principia,"  gives  the  rule  for  belt- 
widths:  "Multiply  36000  by  the  number  of  horse  -  powers ; 
divide  the  amount  by  the  number  of  feet  the  belt  travels  per 
minute  ;  divide  this  quotient  by  the  number  of  feet  in  length  of 
belt  contact  with  the  smaller  drum  or  pulley,  and  divide  this 
by  6 :  the  result  is  the  required  width  of  belt  in  inches." 


VI  PREFACE. 

Professor  Reuleaux  offers  the  formula  b  =  18  \/P,  b  represent- 
ing the  width  of  the  belt  in  millimetres  and  P  the  force  in 
kilograms  transmitted  by  the  belt. 

Unwin,  in  "  Elements  of  Machine  Design,"  gives  the  formula 

2  P 
/3  =  —  in  which  ft  is  the   belt-width  in  inches,  P  the  force 

transmitted  in  pounds,  and  f  the  safe  working  tension  per  inch 
of  width,  which  he  takes  at  70  pounds  for  a  belt  -fa  of  an  inch 
thick.  The  formula  is  to  be  used  only  when  the  belt  embraces 
about  0.4  of  the  smaller  pulley-circumference. 

In  Nystrom's  Mechanics  we  find  b  =  -~- ,  b  denoting  the 

belt  width  in  inches,  H  the  horse-power  transmitted,  d  the 
diameter  of  the  smaller  pulley  in  inches,  and  a  the  number  of 
degrees  occupied  by  the  belt  on  the  circumference  of  the 
smaller  pulley. 

Let  us  now  assume  an  example  which  will  serve  to  determine 
the  variations  in  the  results  of  calculations  from  the  above 
rules  and  formulas.  Suppose  we  wish  to  determine  the  proper 
width  for  a  belt  which  will  transmit  a  force  of  25  horse-power; 
the  smaller  pulley  having  a  diameter  of  5  feet  =  60  inches,  and 
the  velocity  being  10  feet  per  second  =  600  feet  per  minute. 
The  belt  embraces  0.4  of  the  pulley-circumference  =  0.4  x  15.7 
=  6.28  feet  =  360  x  0.4  =  144  degrees.  For  the  force  trans- 
mitted, in  pounds,  we  have  P  —  —  —  =  1375  pounds. 

With  these  quantities  as  data,  Arnold's  rule,  given  above,  gives 

36000  x  25 

us  for   our  required   belt-width  ^-  —  -  =  39.8  inches. 

600  x  6.28  x  6 

If  we  divide  the  force  1375  pounds  by  2.2,  we  obtain  625  kilo- 
grams, and  Reuleaux's  formula  gives  b  =  18  1/625  —  45°  milli- 
metres =  450  x  0.04  =  1 8  inches.  From  Unwin's  formula  we 


PREFACE.  Vli 

obtain  ft  = —  =  39.3  inches,  and  from   the  formula  of 

*7  ^OO   X    2  ^\ 

Nystrom  b  =  -2 =21.7  inches.     Haswell  in  his  "  Engi- 

60  x  144 

neer's  and  Mechanic's  Pocket-book,"  gives  a  rule  by  which  our 
belt-width  would  be  42  inches.  Summing  up  our  results  will 
show  that,  for  the  same  belt,  under  the  same  circumstances, 
the  width  is  according  to  the  authorities  named  as  follows : 

Haswell 42     inches. 

Arnold 39.8         " 

Unwin    39.3 

Nystrom 21.7 

Reuleaux 18 

Of  these  different  values  the  greatest  is  2j  times  the  least. 
Probably  Arnold,  Haswell,  and  Nystrom  are  in  use  in  our  shops 
more  than  the  others,  and  these  give  results,  for  the  belt-width 
in  question,  differing  from  each  other  by  more  than  20  inches. 
According  to  a  list  of  prices  for  double,  white-oak  tanned  belt- 
ing, which  is  before  me,  the  difference  in  cost  for  the  above-cal- 
culated belt,  supposed  to  be  double  and  100  feet  long,  between 
Nystrom  and  Haswell  would  be  six  hundred  and  sixteen  dol- 
lars, to  say  nothing  of  the  difference  in  the  cost  of  the  pulleys, 
shafts,  etc. 

These  great  differences  between  the  results  from  the  rules  of 
different  authors  are  apparently  due  to  the  difference  of  opinion 
concerning  the  value  of  the  coefficient  of  friction,  which  is 
taken  all  the  way  from  0.22  to  0.40,  and  to  the  fact  that  each 
writer  on  the  subject  has  striven  to  obtain  simple  rather  than 
accurate  rules.  At  best  we  are  dealing  with  an  uncertain  ma- 
terial when  we  attempt  to  deduce  rules  for  the  strength  of 
leather  belts,  and  if  the  elements  of  belt-thickness,  method  of 


Vlll  PREFACE. 

lacing  or  fastening,  etc.,  are  entirely  or  partially  neglected,  the 
uncertainty  of  accurate  results  must  be  very  greatly  increased. 
In  the  matter  of  joint-fastening  alone,  a  glance  at  the  table  on 
page  no  will  show  that  a  2o-inch  leather  belt  ^  inch  thick  run- 
ning over  two  equal  cast-iron  pulleys  will  transmit  a  force  of  over 
1800  pounds  with  a  riveted  joint  or  1250  pounds  when  fastened 
with  a  double  raw-hide  lacing,  while  with  a  single  leather-lac- 
ing the  same  belt  will  transmit  but  976  pounds.  In  other 
words,  to  transmit  a  force  of  1000  pounds  over  two  equal  cast- 
iron  pulleys  by  means  of  a  leather  belt  -£%  inch  thick  we  will 
need  a  belt-width  of  12  inches  for  riveted  joint,  16  inches  for 
double  raw-hide  lacing,  and  22  inches  for  single  leather-lacing. 
I  believe  that  it  is  utterly  impossible  for  any  man  to  write  an 
entirely  simple  work  on  the  subject  of  belting,  which  will  be  of 
any  practical  use  to  the  mechanical  world.  The  subject  is 
complicated  by  difficulties  far  greater  than  are  ordinarily  met 
with  in  dealing  with  mechanical  questions,  and  to  attempt  to 
simplify  it  beyond  a  reasonable  limit  is  simply  to  omit  certain 
necessary  considerations,  and  thereby  render  the  investigation 
worthless.  My  object  in  writing  this  work  on  belts  and  pulleys 
is,  therefore,  to  present  to  the  mechanical  public  a  small  yet 
comprehensive  and,  above  all,  an  accurate  book  on  the  subject. 
I  have  constantly  endeavored  to  have  due  regard  for  simplicity, 
yet  when  I  have  found  it  necessary  to  sacrifice  either  simplicity 
or  accuracy,  I  have  invariably  chosen  the  former.  All  measure- 
ments and  dimensions  are  given  in  English  units  in  order  to 
avoid  the  confusion  sometimes  resulting  from  the  use  of  the 
Metric  System,  and  I  have  endeavored  by  numerous  simple  ex- 
amples throughout  the  book  to  fully  illustrate  the  use  of  the 
various  rules  and  formulas.  In  translating  the  part  devoted  to 
metallic  cables  from  Reuleaux,  the  formulas  and  tables  have 


PREFACE.  IX 

been  transformed  from  the  metric  system  into  English  measures, 
which  will,  I  trust,  satisfactorily  explain  the  unusual  numbers 
which  have  resulted  in  a  few  instances. 

In  the  hope  that  my  humble  endeavors  to  furnish  accurate 
information  on  the  subject  of  belt-transmission  to  those  whom 
it  may  concern  may  be  in  a  measure,  if  not  entirely,  successful, 
and  trusting  that  in  the  present  instance  I  may  receive  from 
mechanical  men  the  same  generous  support  and  encourage- 
ment that  have  attended  my  previous  efforts  in  the  field  of 
mechanical  literature,  I  present  to  the  public  my  "  Treatise 

on  Belts  and  Pulleys." 

J.  H.  C. 

NEW  YORK,  May  i,  1885. 


TABLE    OF   CONTENTS. 


SECTION   I. 

PAGE 

Introduction — Absence  of  early  Mechanical  Records — Uncertain 
Origin  of  the  Belt  and  Pulley — Probable  Origin I 

SECTION    II. 

Fundamental  Principles — Direction  of  Rotation — Relations  be- 
tween Circumference,  Diameter,  and  Radius — Velocity — Revo- 
lutions— Power — Horse  power 9 

SECTION    III. 

Rules  for  the  Proper  Disposition  of  Pulleys — Axes  which  coin- 
cide geometrically — Parallel  Axes — Axes  which  intersect  each 
other — Axes  which  cross  without  intersecting 28 

SECTION    IV. 

Transmissions  by  Belts  without  Guides — Half-crossed  Belt — 
Conditions  necessary  for  maintaining  the  Belt  on  the  Pul- 
leys— Distance  between  Pulleys 29 

SECTION   V. 

Transmissions  by  Belts  with  Pulley-guides — Half-crossed  Belt 
with  Pulley  guide — Half-crossed  Belt  with  Movable  Pulley- 
guide — General  Case  of  Crossed  Arbors — Arbors  at  Right  An- 
gles   32 


Xll  TABLE   OF  CONTENTS. 

SECTION   VI. 

PAGE 

Length  of  Belts — Open  Belt — Open  Belt,  approximate  Formula — 
Crossed  Belt — Belts  with  Guides  and  intricately  arranged 45 

SECTION   VII. 

Speed  cones — Stepped  Cones — Open  Belt — Crossed  Belt — Graph- 
ical Method — Continuous-speed  Cones 51 

SECTION   VIII. 

Materials  used  for  Belting — Leather — Vulcani/ed  Rubber — Intes- 
tines of  Animals — Rawhide — Hemp  and  Flax — Leather  and 
Metallic  Wire 65 

SECTION   IX. 

Lacing  and  other  Modes  of  Fastening — Shortening — Single  and 
Double  Lacing — Belt-hooks — Cleat-fastening 68 

SECTION   X. 

Strength  of  Leather  Belts — Resistance  to  Slipping — Coefficient  of 
Friction — Tensions  on  Belts — Breaking-strength — Width  for 
Different  Kinds  of  Fastening — Width  necessary  to  transmit 
certain  Powers 75 

SECTION    XI. 

Leather  Belts  over  Leather-covered  Pulleys— Coefficient  of  Fric- 
tion— Tensions — Width  for  Different  Kinds  of  Fastening- 
Width  necessary  to  transmit  certain  Powers 115 

SECTION    XII. 

Vulcanized  Rubber  Belts — Number  of  Layers  of  Duck — Thick- 
ness— Breaking-strength — Coefficient  of  Friction — Width  for 
Different  Kinds  of  Fastening — Width  necessary  to  transmit 
certain  Powers— Rubber  Belts  over  Leather-  and  Rubber  cov- 
ered Pulleys 14.0 


TABLE   OF  CONTENTS.  xiii 

SECTION   XIII. 

PAGE 

Rim,  Nave,  and  Fixing-keys  for  Pulleys — Rounding  of  the  Rim 
— Flanged  Rim — Rim  of  Pulley  for  Belt  with  Circular  Cross- 
section — Split  Pulleys—  Approximate  Weight  of  Pulleys 159 

SECTION   XIV. 

Arms  of  Pulleys — Oval  Cross-sections  —  Number  of  Arms  — 
Strength  of  Arm  s — Straight  Arms — Single  and  Double  Curved 
Arms 166 

SECTION   XV. 

Shafts — Safe  Shearing  Stress — Steel — Wrought-iron — Cast-iron 
— Diameter  necessary  to  transmit  certain  Powers 171 

SECTION   XVI. 

The  Tightening-pulley — Fast  and  Loose  Pulleys — Reversing  by 
m?ans  of  Fast  and  Loose  Pulleys — Fast  and  Loose  Pulleys  for 
Belts  with  Circular  Cross-sections 179 

SECTION   XVII. 

Rope  belts — Tension  almost  entirely  due  to  the  Weight — Pulley 
for  several  Rope-belts — Proper  diameters  for  Rope-belts — Di- 
ameters of  Pulleys  for  Rope-belts 185 

SECTION   XVIII. 

Jointed  Chain-belts — Rouiller's  Chain-belt — Metallic  Belt  of  Go- 
din — Jointed  Chain-belt  of  Clissold — Coefficient  of  Friction — 
Dimensions '. 192 

SECTION   XIX. 

Tensions  of  Metallic  Cables — Number  of  Strands  and  Wire»»— Co- 
efficient of  Friction 196 


xiv  TABLE    OF  CONTENTS. 

SECTION   XX. 

PAGE 

Calculation  of  Diameters  of  Cables — Formulas  and  Tables  of  Di- 
ameters of  Cables  for  Different  Numbers  of  Wires 200 

SECTION   XXI. 

Deflections  in  the  Cable  of  a  Horizontal  Transmission — Deflec- 
tion of  Cable  in  Motion — Deflection  in  a  State  of  Repose — De- 
flection in  the  Driving  and  Driven  Parts 207 

SECTION    XXII. 

Transmission  by  Cable  with  Increased  Tension — Increased  Di- 
ameters of  Cable  and  Wires 212 

SECTION   XXIII. 

Transmission  by  Inclined  Cable — Tensions  in  Inclined  Cables — 
Deflections — Height  above  the  Ground 217 

SECTION   XXIV. 

Method  of  Tracing  the  Curves  of  Cables — Approximately  Para- 
bolic Curves 221 

SECTION   XXV. 

Transmission  by  Cable  with  Pulleys  near  together — Small  Value 
oi  Si 222 

SECTION   XXVL 

Rim  of  Cable-pulleys — Single  Cable — Several  Cables  upon  one 
Pulley 223 

SECTION   XXVII. 

Arms  and  Nave  of  Cable-pulleys — Number  of  Arms — Oval  Cross- 
sections — Flanged  Cross-sections — Straight  Arms — Curved 
Arms — Reserve  Cables 226 


TABLE   OF  CONTENTS.  XV 


\ 


SECTION   XXVIII. 

PACK 

Pulley-supports  and  Intermediate  Pulleys — Stations  at  the  Ex- 
tremities— Intermediate  Stations — Changing  the  Direction  of 
the  Cable 230 

SECTION    XXIX. 

Dimensions  of  Pulley  supports — Ratio  between  the  Radius  of 
the  Pulley-support  and  Diameter  of  the  Wires 234 

SECTION   XXX. 
Pressure  upon  Axes  of  Pulley  supports — Weight  of  Large  Pulleys  235 

SECTION   XXXI. 

Station  Pillars — Brick  and  Stone  Piers — Pedestals — Two  Pulleys 
side  by  side 238 

APPENDIX   I. 

Experiments  for  determining  various  Coefficients  of  Friction — 
Leather  over  Cast-iron  Pulleys — Leather  over  Leather-covered 
Pulleys — Vulcanized-rubber  Belts  over  Cast-iron  and  Covered 
Pulleys 243 

APPENDIX   II. 

Special  Applications  of  Principles  of  Belts  and  Pulleys — Devices 
for  changing  Motion  and  Direction  of  Rotation — Increasing 
and  Decreasing  Speeds — Intermittent  Motion — Different  Meth- 
ods of  arranging  Principal  Pulley  and  Shop  Shafts  in  Mills. . .  252 


BELTS  AND  PULLEYS. 


§  I.     Introduction. 

Says  Thomas  Ewbank  in  his  famous  "  Hydraulics  and 
Mechanics  :"  "Tradition  has  scarcely  preserved  a  single 
anecdote  or  circumstance  relating  to  those  meritorious 
men  with  whom  any  of  the  useful  arts  originated  ;  and 
when  in  process  of  time  History  took  her  station  in 
the  temple  of  Science,  her  professors  deemed  it  beneath 
her  dignity  to  record  the  actions  and  lives  of  men  who 
were  merely  inventors  of  machines  or  improvers  of  the 
useful  arts ;  thus  nearly  all  knowledge  of  those  to 
whom  the  world  is  under  the  highest  obligations  has 
perished  forever.  ...  A  description  of  the  foundries 
and  forges  of  India  and  of  Egypt,  of  Babylon  and 
Byzantium,  of  Sidon  and  Carthage  and  Tyre,  would 
have  imparted  to  us  a  more  accurate  and  extensive 
knowledge  of  the  ancients,  of  their  manners  and  cus- 
toms, their  intelligence  and  progress  in  science,  than 
all  the  works  of  their  historians  extant,  and  would 
have  been  of  infinitely  greater  service  to  mankind. 

"  Had  a  narrative  been  preserved  of  all  the  circum- 
stances which  led  to  the  invention  and  early  applica- 
tions of  the  lever,  the  screw,  the  wedge,  pulley,  wheel 


2  BELTS  AND   PULLEYS. 

and  axle,  etc.,  and  of  those  which  contributed  to  the 
discovery  and  working  of  metals,  the  use  and  manage- 
ment of  fire,  agriculture,  spinning  of  thread,  matting  of 
felt,  weaving  of  cloth,  etc.,  it  would  have  been  the 
most  perfect  history  of  our  species — the  most  valuable 
of  earthly  legacies.  Though  such  a  work  might  have 
been  deemed  of  trifling  import  by  philosophers  of  old, 
with  what  intense  interest  would  it  have  been  perused 
by  scientific  men  of  modern  times,  and  what  pure  de- 
light its  examination  would  have  imparted  to  every 
inquisitive  and  intelligent  mind  !" 

Rollin,  writing  of  u  The  Arts  and  Sciences  of  the 
Ancients"  many  years  ago,  finds  fault  with  the  world 
for  neglecting  the  great  inventors  and  admiring  the 
military  heroes  of  antiquity.  "  Of  what  utility  to  us 
at  this  day,"  he  asks,  "  is  either  Nimrod,  Cyrus,  or 
Alexander,  or  their  successors,  who  have  astonished 
mankind  from  time  to  time?  With  all  their  magnifi- 
cence and  vast  designs  they  are  returned  into  nothing 
with  regard  to  us.  They  are  dispersed  like  vapors  and 
have  vanished  like  phantoms.  But  the  inventors  of  the 
arts  and  sciences  labored  for  all  ages.  We  still  enjoy 
the  fruits  of  their  application  and  industry;  they  have 
procured  for  us  all  the  conveniences  of  life ;  they  have 
converted  all  nature  to  our  uses.  Yet  all  our  admira- 
tion turns  generally  on  the  side  of  those  heroes  in 
blood,  while  we  scarce  take  any  notice  of  what  we 
owe  to  the  inventors  of  the  arts" 

In  like  manner,  Robertson,  in  his  work  on  India, 
laments  the  loss  of,  or  rather  absence  of,  early  records 
concerning  the  useful  arts  and  sciences.  He  says  :  "  It 
is  a  cruel  mortification,  in  searching  for  what  is  in- 


INTRODUCTION.  3 

structive  in  the  history  of  past  times,  to  find  the  ex- 
ploits of  conquerors  who  have  desolated  the  earth,  and 
the  freaks  of  tyrants  who  have  rendered  nations  un- 
happy, are  recorded  with  minute  and  often  disgusting 
accuracy ;  while  the  discovery  of  useful  arts  and  the 
progress  of  the  most  beneficial  branches  of  commerce 
are  passed  over  in  silence  and  suffered  to  sink  into  ob- 
livion." 

The  origin,  age,  first  application,  and  use  of  the 
mechanism  known  to  us  as  the  "  endless  belt  and 
pulley"  are  entirely  unknown  ;  as  far  back  into  the 
history  of  the  ancients  as  we  can  see  by  means  of  the 
earliest  mechanical  records,  we  find  the  endless  belt 
running  continuously  around  the  pulley  precisely  as  it 
does  to-day.  We  may  theorize,  and  assume  a  probable 
origin  ;  we  may  bring  up,  in  support  of  our  assumption, 
all  the  reason  and  logical  conclusions  at  man's  dispo- 
sal ;  we  may  even  convince  mankind  that  we  have  cor- 
rectly traced  and  explained  the  path  over  which  the 
mechanism  has  come  down  to  us  from  the  dim  ages  of 
the  past.  But  here  we  must  stop ;  we  can  go  no  far- 
ther :  and  the  fact  will  yet  remain  that  the  real  age  and 
origin  for  which  we  are  searching  are  still  undiscovered 
and  unknown.  If,  however,  we  cannot  know  with 
certainty  the  real  age  and  origin  of  belts  and  pulleys, 
it  is  nevertheless  a  satisfaction  to  us  to  be  able  to  trace 
out,  by  analogy,  by  reason,  and  by  the  known  existence 
of  things  which  must  have  necessitated  the  use  of  pul- 
leys, what  seems  to  us  to  have  been  the  origin,  the 
successive  modifications,  and  the  line  of  improvement 
by  which  this  most  useful  contrivance  has  been  handed 
down  to  us. 


4  BELTS  AND  PULLEYS. 

In  searching  for  an  uncertain  origin  or  beginning  of 
anything,  we  most  naturally  start  by  determining  up- 
on the  very  simplest  and  most  rudimentary  form 
(knowing  that  simplicity  almost  always  precedes  com- 
plexity, and  that  a  thing  must  of  necessity  have  a  skele- 
ton before  it  can  have  a  form),  and  then  strive  to  fix 
upon  its  exodus  from  the  conception  to  the  tangible 
thing  itself.  In  order  then  to  trace  the  growth  of  the 
thing  in  question  from  its  origin  to  its  present  much 
altered  and  improved  form,  we  strive  to  imagine  the 
slightest  possible  change,  in  the  right  direction,  which 
can  be  given  to  the  original.  Having  successfully 
achieved  the  first  transformation  or  alteration,  we  con- 
tinue to  pick  out  each  slight  alteration  and  improve- 
ment in  proper  order,  until  we  have  reached  the  present 
most  improved  form. 

If  we  assume,  as  is  claimed  by  some  writers,  that  the 
mechanism  of  the  belt  and  pulley  was  among  the  first 
mechanical  contrivances  of  primitive  man,  we  must 
search  for  its  origin  among  what  we  judge  to  be  the 
first  necessities  of  the  human  race  and  the  modes  of 
obtaining  these  necessities.  Although  many  claim  that 
the  human  race,  in  the  beginning,  passed  through  a 
fireless  period, — that  men  lived  without  the  use  of  fire 
or  artificial  heat, — we  must  nevertheless  conclude  that 
this  element  was  one  of  the  first  necessities  of  human 
life,  and  that  the  first  effort  made  by  prehistoric  man 
in  the  line  of  invention  was  for  the  purpose  of  produc- 
ing fire.  It  is  very  generally  admitted  that  the  first 
"  fire-machine"  (Reuleaux  concludes  that  this  was  the 
first  machine  of  any  description.  See  Kenedey's  Eng- 
lish translation  u  Kinematics  of  Machinery,"  London, 


INTRODUCTION. 


5 


1876,9.204)  consisted  of  an  upright  piece  of  wood, 
having  one  end  pointed.  This,  fitted  into  a  hollow  in 
another  piece  of  wood  and  being  twirled  rapidly  back- 
wards and  forwards  with  the  hands,  generated  sufficient 
heat  to  set  fire  to  some  small  fragments  of  dry  wood 
or  other  combustible  material  (Fig.  i).  Here  we 
have  the  first  belt  and  pulley — hardly  recognizable,  it 
is  true,  but  none  the  less  the  probable  origin.  The 
upright  piece  of  wood  here  constitutes  the  pulley  and 


FIG.  i. 


FIG.  2. 


the  human  hands  the  belt.  The  first  transformation 
seems  to  have  been  the  substitution  of  a  cord  wound 
several  times  around  the  upright  piece  (as  shown  in 
Fig.  2)  in  place  of  the  direct  application  of  the  hands. 

This  rude  contrivance,  though  it  produced  only  an 
oscillating  motion,  was  used  for  other  purposes  than 
that  of  producing  fire;  the  primitive  drill,  lathe,  pot- 
ter's wheel,  etc.,  were  driven  to  and  fro  in  this  manner, 
the  work  being  done  only  on  the  forward  turn,  and  the 


6  BELTS  AND   PULLEYS. 

backward  turn  serving  only  to  place  the  work  in  such 
a  position  that  the  operation  of  cutting  could  be  again 
continued.  The  change  from  this  contrivance  to  the 
rope  and  pulley  used  for  drawing  water  from  deep 
wells,  and  for  lifting  the  vast  blocks  of  stone,  columns, 
etc.,  used  by  the  ancients  in  building,  was  indeed 
slight,  and  may  reasonably  have  taken  place  not  long 
after  the  first  introduction  of  the  improved  form  of 
"  fire-machine.'* 

For  how  long  a  period  this  oscillating  motion  suf- 
ficed for  the  rough  manufacturing  purposes  of  the  age, 
or  at  just  what  era  in  the  life  of  man  the  change  was 
made  to  the  endless  belt,  which  transformed  the  oscil- 
lating into  a  continuous  rotary  motion,  is  indeed  a 
mystery.  Whole  generations — even  centuries — may 
have  been  needed  to  impress  upon  the  primitive  mind 
the  advantages  of  continuous  rotation  and  to  accom- 
plish the  necessary  change  in  the  mechanism.  It 
seems  most  probable  to  us  that  the  loss  of  time  in- 
curred by  the  useless  backward  motion  in  lathes,  drills, 
etc.,  and  the  natural  desire  on  the  part  of  these  an- 
cient artisans  to  accomplish  more  and  more  work  in 
less  and  less  time,  must  have  led  to  the  adoption  of  the 
two  pulleys  and  the  endless  belt.  Gradually,  very 
gradually,  the  slight  but  all-important  change  was 
made.  Some  early  thinker — now  unknown  even  in 
the  uncertain  histories  of  the  past  ages — connected  the 
loose  and  separated  ends  of  the  single  cord,  passed  the 
now  endless  cord  over  two  cylindrical  sticks,  fitted 
roughly  into  a  frame  to  hold  them  apart,  and  caused 
both  to  rotate  by  turning  one  with  a  crank.  Next 
some  primitive  inventor  obtained  the  friction  neces- 


IN  TR  OD  UC  TION.  7 

sary  for  the  transmission  of  considerable  forces  by- 
winding  the  cord  several  times  around  each  pulley ; 
and  so  in  process  of  time,  in  his  attempts  to  obtain 
and  transmit  greater  powers,  the  man  of  the  ages  long 
since  forgotten  at  last  discarded  the  round  cord  for  the 
broad  flat  band  or  belt  of  the  present  era.  Reuleaux 
says  "the  crossed  belt  appears  to  be  the  alder;"  but  to 
us  it  seems  most  probable  that  the  flat  band  was  first 
used  in  its  simplest  form, — i.e.,  open, — and  that  the 
crossed  belt  was  afterwards  introduced  in  order  to  pre- 
vent (by  its  additional  embracing  of  the  pulleys)  slip- 
ping, and  to  produce  a  rotation  of  the  driven  pulley  in 
a  direction  contrary  to  that  of  the  driver. 

As  to  the  material  of  the  primitive  cord  and  belt,  we 
can  prove  nothing :  it  is,  however,  reasonable  to  sup- 
pose, since  the  skin  of  wild  animals  was  the  easiest 
material  to  obtain,  and  since,  from  the  earliest  reeord.s 
of  history,  skins  have  been  used  for  clothing,  bow- 
strings, etc.,  that  the  material  of  the  primitive  belt 
differed  from  the  leather  of  to-day  only  in  that  it  was 
untanned  and  unfinished,  and  perhaps  taken  from  ar 
different  animal.  Doubtless  the  fixing  together  or 
lacing  of  the  ends  of  belts  was  the  source  of  considera- 
ble difficulty  to  the  ancients,  for  in  all  cases  where 
such  a  belt  could  be  made  to  perform  the  necessary 
work,  round  cords  tied  together  at  the  ends  seem  to 
have  been  used. 

It  is  supposed,  and  very  reasonably,  from  certain 
known  circumstances,  that  the  first  idea  of  continuous 
rotary  motion  which  was  developed  in  the  mind  of 
man  took  the  form  of  an  undershot  water-wheel, 
driven  by  the  current  of  a  stream  or  river.  The  Chi- 


8 


BELTS  AND  PULLEYS. 


nese  have  doubtless  used  these  water-wheels,  for  pur- 
poses of  irrigation  and  drawing  water,  for  many  centu- 
ries, and,  according  to  tradition,  they  were  also  used  at 
an  early  date  in  ancient  Assyria,  Mesopotamia,  and 
other  countries  of  Asia  Minor.  These  pristine  water- 
wheels  consisted  of  a  rough  axle  and  two  or  more  long 
blades,  usually  built  up  of  sticks  and  bamboo,  some- 
times with  rough  buckets  formed  out  of  mud  or  clay. 
It  is  not  at  all  unlikely  that  the  first  attempts  to  con- 


FIG.  3. 


struct  large  pulleys  were  founded  upon  the  principles 
of  construction  seen  in  the  water-wheels,  and  that  the 
pulleys  were  used  without  rims,  as  shown  in  Fig.  3. 
From  the  fact  that  wagon-wheels  with  entire  rims  and 
fellies  are  known  to  have  been  in  existence  in  the 
earliest  Greek  and  Egyptian  times,  we  may  very  fairly 
conclude  that  the  use  of  the  complete  wooden,  if  not 
also  iron,  pulley  reaches  far  back  into  antiquity,  and 
that  its  advent  into  the  world  probably  took  place  not 
long  after  the  discovery  of  the  endless  belt. 

History  informs  us  that  the  ancient  city  of  Nineveh 


FUNDAMENTAL   PRINCIPLES.  9 

was  surrounded  by  a  massive  stone  wall  over  100  feet 
high,  and  that  the  city  was  fortified  with  1500  towers, 
each  200  feet  in  height.  Babylon,  "  the  noblest  city 
ever  built  by  man/'  had  a  fortified  wall  which  reached 
to  the  incredible  height  of  360  feet,  and  her  famous 
hanging-gardens  were  built  of  "  flat  stones  of  amazing 
size."  The  Tower  of  Babel  is  said  to  have  been  "40 
rods  square  at  the  bottom,  and  upwards  of  600  feet 
high."  These  gigantic  structures — supposed  to  have 
been  built  about  the  year  2200  B.C. — could  not  have 
been  erected  without  the  aid  of  strong  ropes  and  pul- 
leys, or  similar  contrivance.  Thus  for  over  four  thou- 
sand years  have  been  known  and  used  successfully  the 
cord  and  pulley  which  we  use  to-day.  For  how  many 
centuries  in  the  unknown  ages  of  the  prehistoric  period 
men  toiled  and  labored  with  their  crude  "  fire-ma- 
chines," perhaps  even  lived  and  died  without  reaching 
that  much  of  "the  machine,"  we  must  leave  for  future 
investigation  and  development  to  decide. 

§  2.     Fundamental  Principles. 

The  mechanism  known  in  modern  mechanics  as  the 
"  endless  belt  and  pulleys"  is,  primarily,  a  device,  the 
object  of  which  is  to  transmit  a  continuous  rotary  mo- 
tion from  one  shaft  or  arbor  to  another  parallel  shaft, 
and  the  first  fundamental  principle  of  the  mechanism 
may  be  clearly  expressed  as  follows:  If  two  drums  or 
pulleys  be  placed  in  certain  positions  relative  to  each 
other,  each  being  allowed  the  motion  of  rotation  about 
its  fixed  axis,  and  no  other,  and  if  an  endless  band  be 
passed  tightly  over  the  circumferences  of  the  pulleys 


IO  BELTS  AND  PULLEYS. 

as  represented  in  Fig.  4 ;  then,  if  a  continuous  rotary 
motion  be  given  to  one  of  the  pulleys,  the  friction  be- 
tween it  and  the  band  will  cause  the  latter  to  move 
around  the  circumference,  and  the  second  pulley  will 


FIG.  4. 


(because  of  the  friction  between  it  and  the  band) 
therefore  be  caused  to  rotate  continuously  about  its 
fixed  axis — that  is,  the  continuous  rotary  motion  of 
the  driving-pulley  will  be  directly  transmitted  through 


FIG.  5. 


the  endless  band  to  the  second  pulley.  In  this  defini- 
tion it  is  presupposed  that  the  friction  between  the 
band  and  pulleys  is  sufficiently  great  to  overcome 
the  resistance  of  the  pulleys ;  otherwise  the  driving- 
pulley  will  simply  slide  around  upon  the  band  without 


FUNDAMENTAL   PRINCIPLES.  II 

causing  it  to  move,  and  consequently  the  second  or 
driven  pulley  will  remain  motionless. 

(a)  Direction  of  Rotation. — Belts  maybe  either  open, 
as  shown  in  Fig.  4,  or  crossed,  as  in  Fig.  5  :  in  the 
former  case  the  two  pulleys  rotate  in  the  same  direc- 
tion, while  in  the  latter  case  the  driven  pulley  rotates 
in  a  direction  contrary  to  that  of  the  driver. 

(/;)  Relations  between  Circumference,  Diameter,  and 
Radius. — The  circumference  C  of  a  circle,  the  diameter 
of  which  is  represented  by  D,  is  given  by  the  expres- 
sion 

C=nD, (I) 

in  which  n  represents  the  constant  quantity  3.14159. 

RULE. — To  determine  the  circumference  of  a  circle 
in  inches  or  feet,  multiply  the  diameter  in  inches  or 
feet  by  the  constant  3.14159. 

Since  the  radius  of  a  circle  is  equal  to  one  half  its 
diameter,  if  we  denote  the  radius  by  R,  we  shall  have 

R  —  — ,  or  D  =.  2R,  and  formula  (i)  becomes  by  sub- 
stitution 

C  =  2nR (2) 

RULE. — To  determine  the  circumference  of  a  circle 
in  inches  or  feet,  multiply  the  radius  in  inches  or  feet 
by  the  constant  2n  =  6.28318. 

From  formula  (i),  by  transposing  the  quantities,  we 
may  write 


12  BELTS  AND  PULLEYS. 

\ 

RULE.  —  To  determine  the  diameter  of  a  circle  in 
inches  or  feet,  divide  the  circumference  in  inches  or 
feet  by  the  constant  3.14159. 

In  a  similar  manner  from  formula  (2)  we  may  obtain 

R  =  —  .  .  f4) 

27T 

RULE.  —  To  determine  the  radius  of  a  circle  in  in- 
ches or  feet,  divide  the  circumference  in  inches  or  feet 
by  the  constant  27t  =  6.28318. 

If  we  let  C  and  C  denote  the  circumference  of  two 
circles,  D  and  D'  ,  R  and  R',  the  respective  diameters 
and  radii,  we  shall  have,  from  formulas  (i)  and  (2), 

C  =  nD  =  2nR,        and         C  =  TtD'  =  2nRr  ; 
and  we  may  write  the  proportions 

C  :  C  ::  nD  :  nDf  ::  2nR 
in  the  form  of  an  equation, 

C        nD        2nR 


which,  by  cancelling  the  equal  constants  in  numerator 
and  denominator,  becomes 

D        R 


RULE.  —  The  ratio  of  the  circumferences  of  any  two 
circles  is  equal  to  the  direct  ratio  of  their  diameters  or 
radii. 


\ 


FUNDAMENTAL   PRINCIPLES.  13 

(V)  Velocity. — The  circumferential  velocities  of  two 
pulleys  which  are  connected  by  one  and  the  same  belt 
(supposing  there  is  no  slipping  of  the  belt  on  either 
pulley)  must  obviously  be  the  same,  each  being  equal 
to  the  velocity  of  the  belt.  For  the  belt  must  unroll 
from  the  driving-pulley  just  as  fast  as  it  is  developed 
from  the  pulley-circumference ;  it  must  also  roll  upon 
the  circumference  of  the  driven  pulley  with  the  same 
velocity,  else  the  belt  would  constantly  tend  to  become 
tighter  on  one  side  and  looser  on  the  other,  and  sliding 
or  rupture  would  necessarily  ensue."* 

The  circumferential  velocity  of  the  driven  pulley  and 
the  velocity  of  the  belt  are  entirely  independent  of  the 
pulley-diameters,  and  depend  solely  upon  the  circum- 
ferential velocity  of  the  driving-pulley.  Thus,  if  the 
circumferential  velocity  of  the  driver  is  10  feet  per 
second,  10  feet  of  circumference,  and  no  more  or  less, 
can  be  developed  per  second  upon  the  belt,  be  the 
driver  ever  so  large  or  ever  so  small.  In  the  same 
manner,  just  10  feet  of  belt  can  roll  per  second  upon 

*  The  tensions  on  the  two  s,ides  (or  parts)  of  the  belt  are  not  the 
same  (as  will  be  seen  farther  on);  consequently  the  circumferential 
velocities  of  the  two  pulleys  are  not  absolutely  the  same.  According 
to  Professor  Reuleaux,  if  v  and  v  denote  the  circumferential  velo- 
cities of  the  two  pulleys,  t  and  T  the  tensions  on  the  two  parts  of  the 
belt,  E  the  coefficient  of  elasticity  o,f  the  belt,  and  S  the  strain  on  the 
driving  part  of  the  belt,  the  true  velocities  will  be  given  by  the  ex- 
t 

pression  —       —  =  .     Reuleaux  says,  "  The  loss  of  velocity  due 

~S 

to  the  sliding  has  for  a  mean  value  about  \  percent;  it  is  accompanied 
by  a  loss  of  work,  which  is  transformed  into,  heat  and  produces  wear 
of  the  belt  and  pulleys." 


14  BELTS  AND  PULLEYS. 

the  circumference  of  the  driven  pulley,  without  refer- 
ence to  its  size  or  diameter. 

(//)  Revolutions.  —  Since  the  circumferential  velocities 
of  any  two  pulleys,  which  are  connected  by  one  and 
the  same  belt,  are  the  same  without  regard  to  the 
diameters  of  the  pulleys,  and  since  the  circumferences 
of  the  two  pulleys  are  directly  proportional  to  their 
diameters  (formula  5)  ;  if  one  of  the  pulleys  has  a 
diameter  equal  to  twice  that  of  the  other,  the  circum- 
ference of  the  former  will  also  be  equal  to  twice  that 
of  the  latter,  and  the  former  will  need  just  twice  as 
much  time  in  which  to  perform  one  entire  revolution 
as  the  latter.  In  other  words,  the  larger  pulley  will 
make  just  one  half  as  many  revolutions  in  a  given  time 
as  the  smaller.  In  a  similar  manner,  if  the  diameter 
of  the  larger  pulley  is  three  or  four  times  that  of  the 
smaller,  the  former  will  need  three  or  four  times  as 
much  time  for  each  revolution  as  will  the  latter,  or  the 
larger  pulley  will  make  only  one  third  or  one  fourth 
the  number  of  revolutions  in  a  given  time  as  the 
smaller.  In  formula,  denoting  by  n  and  ri  the  num- 
bers of  revolution  of  the  two  pulleys,  and  by  C  and  C', 
D  and  D\  and  R  and  R  the  respective  circumferences, 
diameters,  and  radii,  we  shall  have 


— 
C"  D 


RULE.  —  The  ratio  of  the  numbers  of  revolutions  of 
two  pulleys,  which  are  connected  by  one  and  the  same 
belt,  is  equal  to  the  inverse  ratio  of  their  circumfer- 
ences, diameters,  or  radii. 


FUNDAMENTAL   PRINCIPLES  15 

If  we  represent  by  n  the  number  of  revolutions  per 
minute,  by  vm  the  velocity  in  feet  per  minute,  and  by 
Rf  and  Cf,  respectively,  the  radius  and  circumference 
of  the  pulley  in  feet,  we  shall  have  for  the  velocity  the 
expression 

vm  =  27tRfn  =  Cfn (7) 

RULE. — To  determine  the  velocity,  in  feet  per  min- 
ute, with  which  a  pulley  rotates,  multiply  the  circum- 
ference of  the  pulley,  in  feet,  by  the  number  of  revolu- 
tions per  minute. 

If  R  and  C  denote  respectively  the  radius  and  cir- 
cumference-of  the  pulley  in  inches,  we  shall  have,  be- 

r> 

tween  R  and  Rf,  C  and  Cf  the  relations  Rf  =  --  and 

c 

Cf  =  — .     These  values,  substituted  in  formula  (7),  give 


27tRn       Cn 

—  —  =  -^  =  0.5236^.  ...     (8) 


RULE. — To  determine  the  velocity  of  a  pulley  in  feet 
per  minute,  multiply  the  circumference  of  the  pulley  in 
inches  by  the  number  of  revolutions  per  minute,  and 
divide  the  product  by  12,  or  multiply  0.5236  times  the 
radius  in  inches  by  the  number  of  revolutions  per 
minute. 

Let  v  represent  the  velocity  of  the  pulley  in  feet  per 
second ;  we  shall  then  have  the  expression 

7;    =  6ov 


l6  BELTS  AND  PULLEYS. 

and  formula  (8)  becomes,  by  substitution, 

277 Rn       Cn 

6™--   —2~   --7t> 

which  reduces  to 


v  =         =  0.00873^.  .    .    .    (9) 


RULE.  —  To  determine  the  velocity  of  a  pulley  in 
feet  per  second,  multiply  the  circumference  of  the  pul- 
ley in  inches  by  the  number  of  revolutions  per  minute, 
and  divide  the  product  by  720,  or  multiply  0.00873 
times  the  radius  in  inches  by  the  number  of  revolutions 
per  minute. 

If  we  substitute  the  value  vm  =  6ov  in  formula  (7), 
we  shall  obtain  the  expression 


6ov  = 
which  reduces  to 


C  '  n 
v  =  -~-  =  o.iotfRfn.  .     .     .     (10) 


RULE.  —  To  determine  the  velocity  of  a  pulley  in 
feet  per  second,  multiply  by  the  circumference  of  the 
pulley  in  feet  by  the  number  of  revolutions  per 
minute,  and  divide  the  product  by  60;  or  multiply 
0.1047  times  the  radius  in  feet  by  the  number  of  re- 
volutions per  minute. 


FUNDAMENTAL   PRINCIPLES.  IJ 

By  transposing  formula  (7),  we  may  obtain,  for  the 
number  of  revolutions  per  minute,  the  formula 

vm         vm 

n  =  —  --  —  —-.      .     .     .     .      (II) 
2nRf        Cf 

RULE.  —  To  determine  the  number  of  revolutions  per 
minute  with  which  a  pulley  turns,  divide  the  velocity 
of  the  pulley  in  feet  per  minute  by  the  pulley-circum- 
ference in  feet. 

In  a  similar  manner,  by  transposing  formulas  (8),  (9), 
and  (10)  we  may  obtain  the  following  formulas  for  the- 
number  of  revolutions  per  minute  : 

n  =  --^  =  —  ^  (12) 

C        0.5236^ 

RULE.  —  To  determine  the  number  of  revolutions  per 
minute,  divide  12  times  the  velocity  in  feet  per  minute 
by  the  circumference  of  the  pulley  in  inches  ',  or  divide 
the  velocity  in  feet  per  minute  by  0.5236  times  the 
radius  of  the  pulley  in  inches. 

J2OV  V 

/M     _     '  _  I    T  O  \ 

fi    —         7^       —    —         '::        7^.  .       •        •       I  I  s  I 

C         0.008737? 

RULE.  —  To  determine  the  number  of  revolutions  per 
minute,  divide  720  times  the  velocity  in  feet  per  second 
by  the  circumference  of  the  pulley  in  inches,  or  divide 
the  velocity  in  feet  per  second  by  0.00873  times  the 
radius  of  the  pulley  in  inches. 


, 

n  ~  -FT-  —  —      —  ^-  .....     (14) 
Cf 


18  BELTS  AND  PULLEYS. 

RULE.  —  To  determine  the  number  of  revolutions  per 
minute,  divide  60  times  the  velocity  in  feet  per  second 
by  the  circumference  of  the  pulley  in  feet,  or  divide 
the  velocity  in  feet  per  second  by  0.1047  times  the 
radius  of  the  pulley  in  feet. 

The  numbers  of  revolutions  per  minute  of  two  or 
more  pulleys,  which  are  fixed  upon  one  and  the  same 
shaft,  must  plainly  be  the  same,  for  the  shaft  at  each 
revolution  will  carry  each  and  all  of  the  pulleys  just 
once  around  without  reference  to  the  diameters  of  the 
pulleys.  If,  therefore,  we  denote  by  n  the  common 
number  of  revolutions,  and  by  v  and  vf  the  circum- 
ferential velocities  of  two  pulleys,  which  are  fixed  upon 
one  and  the  same  shaft,  we  shall  have,  from  formula 
(9),  the  equations 

Cn 

•v  =  -  -  —  0.008  7  T>Rny 
720 

J~>f 

and  v'  —  --  =  o.oo873^X 

C,  R,  C',  and  Rf  denoting  respectively  the  circumfer- 
ences and  radii  of  the  two  pulleys.  From  these  two 
equations  we  may  write  the  proportion 

/~*       /~*t 

v  :  vf  ::  —  :  -  -  ::  0.00873^/2  :  0.00873^^. 
720    720 

By  cancelling  out  the  equivalent  quantities,  and  writ- 
ing the  proportion  in  the  form  of  an  equation,  we  have 


C 


FUNDAMENTAL  PItlNCIPLES.  IQ 

-The  ratio  of  the  velocities  of  two  pulleys 
which  are  fixed  upon  one  and  the  same  shaft  is  equal 
to  the  direct  ratio  of  the  pulley  circumferences,  radii, 
or  diameters. 

(e)  Power. — By  the  power  of  a  pulley  we  mean  the 
force  with  which  the  circumference  of  the  pulley  turns: 
it  is  equal  to  that  force  which,  if  applied  to  the  pulley- 
circumference  in  a  direction  opposite  to  that  in  which 
the  pulley  rotates,  would  be  just  sufficient  to  stop  the 


FIG.  6. 

motion  of  the  pulley.  The  powers  of  two  pulleys 
which  are  connected  by  one  and  the  same  belt  are 
equal ;  for  the  driving-pulley  transmits  all  its  circum- 
ferential force  to  the  belt,  and  the  belt  in  turn  trans 
mits  the  same  force  to  the  driven  pulley  (less  a  very 
slight  amount  which  is  consumed  in  the  stretching  of 
the  belt). 

Let  the  circles  of  which  the  radii  are  R,  R ,  r,  and  A 
(Fig.  6)  represent  four  pulleys,  connected  by  belts  as 
shown  in  the  figure,  A  being  the  driving-pulley  and  R 


20 


BELTS  AND  PULLEYS. 


and  r  being  fixed  upon  one  and  the  same  shaft.  The 
power  P  of  the  driving-pulley  is  transmitted  directly  to 
the  pulley  R  through  the  belt  xy.  We  may  consider 
the  imaginary  line  abc  as  a  simple  lever,  the  fulcrum  of 
which  is  at  the  point  a,  and  the  arms  of  which  are  ac 
and  ab.  If  now  we  let  P  represent  the  power  of  the 
pulley  r,  which  is  transmitted  directly  to  the  pulley  R! 
through  the  belt  x 'y' ,  we  shall  have,  from  the  principles 
of  the  simple  lever,  the  relation 


or 


PR  =  Pr, 
P         r_ 
P  ~~~  ~R° 


(16) 


RULE. — The    ratio   of   the   powers   of   two   pulleys 
which  are  fixed  upon  one  and  the  same  shaft  is  equal 


FIG.  7. 


to  the  inverse  ratio  of  the  pulley-radii  (diameters  or 
circumferences). 

Let  the  circles  of  Fig.  7  represent  a  number  of 
pulleys,  connected  by  belts  as  shown  in  the  figure,  and 
together  constituting  a  "  pulley-train."  Let  A  be  the 
driving-pulley,  and  let  r'f  be  arranged  to  lift  the  weight 


FUNDAMENTAL   PRINCIPLES.  21 

Pfl  by  means  of  a  cord  wound  around  its  circumfer- 
ence, as  shown  in  the  figure.  From  formula  (16)  we 
shall  have  the  expression 

P        r  PR 

P  =  R       or       p  =  T-- 

Also,  we  shall  have 

P.  -  ?L  P"  -  FRr 

P'  ~  R  '  r'  ' 

Substituting,  in  the  last-found  equation,  the  value  of 
P  determined  above,  gives 

_  PRR_ 
rr' 

From  formula  (16)  again  we  may  write  the  equation 

T)'f              „//  P"  Rff 
' nr                   n'ff    ____  . 

P"  —  Rn  r"    J 

and  by  substituting  in  this  the  last-found  value  of  />/;, 
we  shall  finally  obtain  the  formula 

_„  =  PRR'R" 


rr'r" 

P'!frr'rfr 
Then,  inversely,          P  =  RR'Rfr (l8) 

RULE. — To  determine   the  power  of   an  increasing 
pulley-train  (one  in  which  the  powers  of   the  pulleys  . 


22  BELTS  AND  PULLEYS. 

constantly  increase  from  the  driver),  multiply  the 
power  of  the  driver  by  the  continued  product  of  all 
the  larger  pulley-radii  (diameters  or  circumferences) 
except  that  of  the  driver,  and  divide  the  result  by 
the  continued  product  of  all  the  smaller  pulley-radii 
(diameters  or  circumferences)  except  that  of  the 
driver.  To  determine  the  power  of  a  decreasing  pulley- 
train  (one  in  which  the  powers  of  the  pulleys  con- 
stantly decrease  from  the  driver),*  multiply  the  power 
of  the  driver  by  the  continued  product  of  all  the 
smaller  pulley-radii  (diameters  or  circumferences)  ex- 
cept that  of  the  driver,  and  divide  the  result  by  the 
continued  product  of  all  the  larger  pulley-radii  (diame- 
ters or  circumferences)  except  that  of  the  driver. 

From  formula  (15)  we  know  that  the  circumferential 
velocities  of  two  pulleys  which  are  fixed  upon  one  and 
the  same  shaft  vary  directly  as  the  pulley  radii, 
diameters,  or  circumferences.  We  may  therefore  ob- 
tain, by  combining  formulas  (15)  and  (16)  and  denoting 
the  circumferential  velocities  of  the  pulleys  R  and  r 
(Fig.  6)  oy  Kand  v  respectively, 


('9) 


RULE.  —  The   ratio  of   the    powers    of   two    pulleys 
which  are  fixed  upon  one  and  the  same  shaft  is  equal 

*  If  the  pulley-train  represented  in  Fig.  7  were  a  decreasing  in- 
stead of  an  increasing  train,  the  "direction"  of  the  train  would  be 
reversed.  That  is,  the  pulley  /v>"  would  be  the  driver  and  the  pulley 
A  the  one  which  lifts  the  weight. 


FUNDAMENTAL    PRINCIPLES.  23 

to  the  inverse  ratio  of  the  circumferential  velocities  of 
the  pulleys. 

A  glance  at  formula  (19)  will  show  that  the  increased 
power  which  we  obtain  by  means  of  an  increasing  pul- 
ley-train necessitates  a  loss  of  time  corresponding  to 
the  gain  in  power.  For  since  the  power  varies  in- 
versely as  the  velocity,  if  we  increase  the  power  two, 
three,  or  four  fold  we  necessarily  decrease  the  velocity 
two,  three,  or  four  fold  also.  Thus,  if  by  means  of  the 
train  represented  in  Fig.  7  we  can  lift  a  weight  of  1000 
pounds  with  a  circumferential  force  on  the  driving- 
pulley  amounting  to  say  200  pounds  only,  we  will  need 
just  ^y  =  5  times  as  much  time  as  if  we  apply  the 
force  of  1000  pounds  directly  to  the  pulley  which  lifts 
the  weight.  Nevertheless  there  is  a  real  gain  repre- 
sented in  the  increasing  pulley-train ;  because,  without 
it  or  a  similar  contrivance,  we  might  tug,  with  our  200 
pounds  of  power,  for  a  lifetime,  and  still  be  unable  to 
lift  the  1000  pound  weight  one  inch  from  its  resting- 
place. 

(/")  Horse-power. — The  term  "  horse-power,"  as  com- 
monly used,  is  equivalent  to  33,000  foot-pounds:  it  is 
that  amount  of  force  or  power  which  will  lift  a  weight 
of  33,000  pounds  one  foot  high  in  one  minute,  or  a 
weight  of  one  pound  33,000  feet  high  in  one  minute. 
If  we  represent  the  horse-power  of  a  pulley  by  77,  and 
the  circumferential  force  or  power  in  pounds  byP,  then 
H  X  33,000  pounds  lifted  one  foot  high  per  minute 
will  represent  the  power  of  the  pulley.  If  therefore 
we  denote  by  vm  the  circumferential  velocity  of  the 
pulley  in  feet/^r  minute,  wre  shall  have,  for  the  power 
in  pounds,  the  expression 


24  BELTS  AND  PULLEYS. 


*  And  inversely,         H  =  -^-.  (21) 

33000 

RULE.  —  To  determine  the  power  of  a  pulley  in 
pounds,  divide  33000  times  the  horse-power  by  the  cir- 
cumferential velocity  of  the  pulley  in  feet  per  minute  : 
to  determine  the  horse-power,  multiply  the  power  of 
the  pulley  in  pounds  by  the  circumferential  velocity  in 
feetjfor  minute  and  divide  the  product  by  33000. 

If  v  denote  the  circumferential  velocity  of  the  pul- 
ley in  feet  per  second,  we  shall  have  the  relation 
vm  =  6oz>,  and  formula  (20)  becomes,  by  substitution, 

p=  3300Q//" 

6ov     ' 


P=.    ...... 

V 

Pv 
Inversely,  #==-—  ........     (23) 

RULE.  —  To  determine  the  power  of  a  pulley  in 
pounds,  divide  550  times  the  horse-power  by  the  cir- 
cumferential velocity  in  feet  per  second  ;  to  determine 
the  horse-power,  multiply  the  power  of  the  pulley  in 
pounds  by  the  circumferential  velocity  in  feet  per 
second,  and  divide  the  product  by  550. 

The  size  of  a  pulley  is  usually  given  in  terms  of  its 
diameter:  thus  a  "36-inch  pulley"  is  a  pulley  the 


FUNDAMENTAL   PRINCIPLES.  2$ 

diameter  of  which  is  36  inches;  a  "4-foot  pulley"  is 
one  the  diameter  of  which  is  4  feet. 

Example  i. — The  diameter  of  a  pulley  is  10  inches; 
it  is  required  to  find  the  circumference.     From  formula, 
(i)  we  have  C  =  nD  =  3.14159  X  10  or  C  =  31.4159". 

Also  we   have  R  —  --  =  5",  and    formula   (2)  gives 

2 

C  =  2?tR  =  6.28318  X  5  or  C  —  31.4159". 

Example  2. — The  circumference  of  a  pulley  is  C  = 
314.159";  it  is  required  to  find  the  diameter.  We  have, 

from  formula  (3),  D  =  ~  =  m£59  =  100". 

n        3.14159 

Example  3. — The  diameters  of  two  pulleys,  which 
are  connected  by  one  and  the  same  belt,  are  D  =  30" 
and  Dr  =  10"  ;  the  larger  pulley  makes  n  •=.  120  revo- 
lutions per  minute.  It  is  required  to  determine  the 
number  of  revolutions  per  minute  of  the  smaller  pul- 

n        D'       120        10 

ley.     rrom  formula  (6)  we  have  —  =  -^  or  — r  =  — • 

n         D         n          30 

30  X  120 

rrom  this,  n  — =  360. 

10 

Example  4. — A  pulley,  the  radius  of  which  is  2  feet, 
makes  100  revolutions  per  minute;  it  is  required  to  de- 
termine the  circumferential  velocity  in  feet  per  minute. 
We  have,  from  formula  (7),  vm  =  27tRfn,  or  vm  — 
6.28318  X  2  X  100  =  1256.6. 

Example  5. — The  radius  of  a  pulley  is  24  inches,  and 
the  number  of  revolutions  per  minute  100;  it  is  re- 
quired to  determine  the  circumferential  velocity  of  the 
pulley  in  feet  per  minute.  From  formula  (8)  we  have 

2nRn                  6.28318  X  24  X  loo 
vm  —  -    —  or  vm  --  =  1256.6. 

12  12 


26  BELTS  AND  PULLEYS. 

Example  6. — The  radius  of  a  pulley  is  24  inches  and 
the  number  of  revolutions  per  minute  loo ;  it  is  re- 
quired to  determine  the  circumferential  velocity  of  the 
pulley  in  feet  per  second.  Formula  (9)  becomes,  by  sub- 
stituting the  numerical  data,  v  =  0.00873  X  24  X  100, 
or  v  —  20.95. 

Example  7. — The  circumferential  velocity  of  a  pul- 
ley is  1256.6  feet  per  minute,  and  the  radius  2  feet ;  it 
is  required  to  find  the  number  of  revolutions  per 

v 
minute.      From    formula   (11)    we   have  n  —  — —^  = 

1256.6 
6.28318  X  2  = 

Example  8. — It  is  required  to  determine  the  number 
of  revolutions  per  minute  of  a  pulley  of  which  the 
radius  is  24",  and  the  circumferential  velocity,  in 
feet  per  second.  20.95.  From  formula  (13)  we  have 

v       '  _          20.95 
'  0.00873^  ~  0.00873  X  24  = 

Example  9. — A  shaft  which  makes  100  revolutions 
per  minute  bears  two  pulleys  of  which  the  radii  are 
R  =  36  inches  and  R'  =  24  inches ;  it  is  required  to 
determine  the  circumferential  velocities  of  the  two 
pulleys  in  feet/^r  second.  From  formula  (9)  we  have, 
for  the  circumferential  velocity  of  the  pulley  R ',  v'  = 
o  00873  X  24  X  IOO  =  20.95  feet  per  second,  and  from 

v  36  20.95  X  36 

formula  (lO  we  have  -      -  =  — ,  or  v  =  —         —  = 
20.95       24  24 

31.425  test  per  second. 

Example  10. — In  an  increasing  pulley-train  we  have 
the  following  data:  Power  of  the  driving-pulley  P=.  IOO 
pounds,  radii  of  the  pulleys  (of  which  there  are  six 


FUNDAMENTAL  PRINCIPLES.  2/ 

besides  the  driver,  and  arranged  as  shown  in  Fig.  7), 
R  :  =  R'  =  R"  =  36"  and  r  =  V  =  r",=  12"  ;  it  is 
required  to  determine  the  power  of  the  pulley-train. 
By  substituting  the  above  values  in  formula  (17)  we 

100  X  36  X  3^  X  36 
obtain  P"  =  -  -  =  2700  pounds. 

12   X   12   X   12 

Example  n.  —  Suppose  the  circumferential  velocity 
of  the  driving-pulley  in  Example  10  is  1200  feet  per 
minute  ;  it  is  required  to  determine  the  circumferential 
velocity  of  the  pulley  r".  From  formula  (19)  we 

TOO 

From  this,  «^  = 


1200  X    TOO 

-  =  44.44  ieet  /><?r  minute. 
2700 

Example  12.  —  Required  to  determine  the  power  of  a 
pulley  which  transmits  60  horse-power  at  a  circumfer- 
ential velocity  of  10  {QQ^.  per  second.  From  formula  (22) 


D  550  X  60 

we  have  P  =  -       -  or  /*  ==  -^  —  =  3300  pounds. 

z/  10 

Example  13.  —  The  circumferential  force  or  power  of 
a  pulley  is  3300  pounds,  and  the  velocity  10  feet  per 
second;  it  is  required  to  determine  the  horse-power 
transmitted  by  the  pulley.  Formula  (23)  gives  //  = 
Pv  3300  X  10 


550  550 


=  60. 


28  BELTS  AND  PULLEYS. 


§  3.     Rules  for  the  Proper  Disposition  of  Pulleys.* 

The  axes  of  two  pulleys  which  are  connected  by  one 
and  the  same  belt  may  bear  to  each  other  the  follow- 
ing relations : 

1.  They  may  coincide  geometrically. 

2.  They  may  be  parallel. 

3.  They  may  intersect  each  other. 

4.  They  may  cross,  without  being  in  the  same  plane. 

In  these  different  cases  the  belt  passes  from  the 
driving  to  the  driven  pulley,  either  directly  or  by 
means  of  intermediate  pulleys  or  pulley-guides.  It  is, 
first  of  all,  indispensable  that  the  pulleys  be  placed  in 
such  a  manner  that  the  belt  shall  maintain  its  proper 
position  upon  both  pulleys  without  running  off  or 
compelling  recourse  to  special  guides.  The  geometric 
disposition  of  the  pulleys  by  which  this  condition  may 
be  fulfilled  is  called  the  "  arrangement"  of  the  belt. 

The  preceding  condition  will  be  satisfied  if  the  pul- 
leys are  so  placed  with  reference  to  each  other  that,  for 
each  of  them,  the  median  line  of  that  portion  of  the  belt 
which  runs  toward  the  pulley  is  in  the  middle  plane  of 
the  pulley. 

In  pulleys  which  have  rounded  fellies  (see  §  13)  slight 
variations  from  this  rule  (from  £°  to  f°)  may  be  ad- 
missible. 

*  §§  3,  4,  and  5  from  Reuleaux. 


TRANSMISSIONS  BY   BELTS  WITHOUT  GUIDES.    29 


§4.     Transmissions  by  Belts  without  Guides. 

The  simplest  and  most  common  arrangements  of 
pulleys  are  those  in  which  the  belt  passes  directly  from 
one  pulley  to  the  other  without  guides  of  any  kind  ; 
the  simplest  of  these  dispositions,  which  corresponds 
to  the  case  in  which  the  axes  of  the  pulleys  are  parallel, 
is  represented  in  Fig.  8.  In  the  left-hand  figure  the 
belt  is  open,  and  the  pulleys  rotate  in  the  same  direc- 


FIG.  8. 

tion :  in  the  figure  on  the  right  the  belt  is  crossed  and 
the  pulleys  rotate  in  opposite  directions.  In  these  two 
arrangements  the  belt  may  run  in  either  direction,  the 
condition  which  prevents  its  running  off  the  pulleys 
being  fulfilled  for  either  direction  of  rotation. 

For  pulleys  the  axes  of  which  coincide  geometrically, 
as  for  those  in  which  the  axes  intersect,  it  is  evidently 
impossible  to  establish  transmission  without  guides. 


BELTS  AND  PULLEYS. 


For  the  case,  however,  in  which  the  axes  cross  without 
being  in  the  same  plane,  belts  without  guides  may  be 
used  with  the  arrangement  of  pulleys  represented  in 
-  9>  which  is  very  frequently  seen  in  practice. 

i 
js 


FIG.  9. 


This  disposition  allows  us  to  dispense  with  all  ex- 
terior guides,  if  we  are  careful  to  place  the  pulleys  in 
such  a  manner  that  the  line  of  intersection  of  their 
middle  planes  shall  be  tangent  to  the  circles  contained  in 


TRANSMISSIONS  BY  BELTS    WITHOUT  GUIDES.    31 

these  planes  at  the  points  in  zvhlch  tJie  belt  leaves  the 
pulleys.  In  Fig.  9,  in  which  a  and  bl  are  these  points, 
the  belt  must  run  in  the  direction  indicated  by  the 
arrows.  If  we  wish  to  run  the  belt  in  a  contrary  direc- 
tion it  is  necessary  to  move  the  pulleys  upon  their 
arbors  until  the  line  of  intersection  of  their  middle- 
planes  becomes  tangent  to  the  circles  at  the  points  al 
and  b.  This  condition  is  fulfilled  when,  with  reference 
to  the  crossing  K  of  the  pulley-axes,  the  new  positions 
occupied  by  the  pulleys  are  found  to  be  symmetrical 
with  the  positions  of  the  pulleys  before  the  change. 

The  transmission  represented  in  Fig.  9  may  be  con- 
sidered as  the  general  solution  of  transmissions  by  belts 
without  guides.  It  gives,  in  fact,  the  transmission  by 
open  belt,  when  the  angle  /3  included  between  the 
middle  planes  of  the  pulleys  is  equal  to  o,  and  the 
transmission  by  crossed  belt  when  this  angle  is  equal 
to  1 80°.  In  all  intermediate  positions  the  belt  is  only 
partially  crossed  :  for  /3  =  90°,  we  have  a  half-crossed 
belt,  for  fi  =  45°  a  crossing  of  one  fourth,  etc. 

In  short,  partially  crossed  belts,  the  tendency  to  run 
off  the  pulleys  is  very  great.  According  to  Redten- 
bacher,  in  order  that  this  accident  may  be  avoided, 
the  distance  between  the  centres  of  the  pulleys  should 
not  be  less  than  twice  the  diameter  of  the  largest 
pulley ;  that  is,  the  angle  of  deviation  of  the  belt  should 
not  exceed  25°.  Moreover,  in  order  that  the  wear  of 
the  belt  may  not  be  excessive,  the  distance  between 
the  centres  of  the  pulleys  should  not  be  less  than 
10  VbD,  b  representing  the  width  of  the  belt  and  D 
the  diameter  of  the  driving-pulley.  It  is  evident  that, 
in  each  particular  case,  it  is  advantageous  to  take,  for 


32  BELTS  AND  PULLEYS. 

the  separation  of  the  pulleys,  the  greater  of  these  two 
values. 


§  5.      Transmissions  by  Belts  with  Pulley-Guides. 

RULE. — In  a  transmission  by  belt  with  pulley-guides, 
in  order  that  the  belt  may  run  properly  upon  the 
pulleys  and  pulley-guides,  the  point  in  which  the  belt 
leaves  each  pulley  must  be  the  point  of  tangency  be- 


FlG. 


FIG.  ii. 


tween  the    pulley  and  the  line  of  intersection  of    its 
middle  plane  with  that  of  the  following  pulley. 

Figs.  10  and  II  represent  transmissions  of  this  kind 
for  pulleys  with  parallel  axes.  In  Fig.  10  the  middle 
planes  of  the  two  pulley-guides  are  tangent  to  the  two 
pulleys  of  transmission  A  and  B,  and  their  common 
diameter  is  equal  to  the  distance  between  the  middle 


TRANSMISSIONS  BY  BELTS  WITH  GUIDES.      33 

planes  of  these  pulleys.  This  disposition  of  pulleys 
permits  of  the  movement  of  the  belt  in  either  direction. 
When,  as  is  most  commonly  the  case,  a  movement  of 
the  belt  in  one  direction  is  sufficient,  we  may  make  use 
of  the  simpler  disposition  of  pulleys  represented  in 
Fig.  II,  in  which  the  axes  of  the  pulley-guides  coincide 
geometrically.  A  and  B  are  the  pulleys  of  transmis- 


FlG.     12. 

sion  ;  the  middle  planes  of  the  pulley-guides  are  par- 
allel, and  are  tangent  respectively  to  the  pulleys  A  and 
B  at  the  points  in  which  the  belt  leaves  the  latter 
pulleys.  The  common  diameter  of  the  pulley-guides 
is  equal  to  the  distance  between  the  middle  planes/ of 
the  pulleys  of  transmission.  As  indicated  in  the  figure, 
the  pulleys  of  transmission  A  and  B  rotate  in  opposite 
directions. 


34 


BELTS  AND  PULLEYS. 


If  we  consider  B  as  a  pulley-guide  (in  which  case  it 
may  run  loose  upon  the  arbor  of  A),  the  two  pulleys 
£7  and  D  may  be  taken  as  pulleys  of  transmission,  and 
fixed  upon  two  separate  arbors,  the  directions  of  which 
are  the  same. 

If  the  pulley-guides  C  and  D  are  placed  between  the 
arbors  of  A  and  B,  as  is  indicated  in  Fig.  12,  they  will 
rotate  in  the  same  direction,  and  may  consequently  be 


FIG.  13. 

fixed  upon  one  and  the  same  arbor.  The  pulleys  of 
transmission  A  and  B  will  also  rotate  in  the  same  direc- 
tion. In  this  case  the  belt  can  move  in  one  direction 
only,  and  remain  properly  upon  the  pulleys  and  guides. 
The  two  pulley-guides  C  and  D  may  be  replaced  by  a 
single  pulley,  provided  it  is  placed  obliquely  so  as  to 
run  on  both  sides  of  the  belt  without  causing  displace- 
ment. 


BY  BELTS   WITH  GUIDES-      35 

Fig.  13  represents  a  transmission  by  belt  for  two 
pulleys,  the  axes  of  which  intersect  each  other.  In 
this  disposition,  which  differs  from  that  of  Fig.  11  only 
in  the  inclination  of  the  axis  of  the  pulley  B,  the 
movement  of  the  belt  can  take  place  only  in  one  direc- 
tion. To  obtain  a  movement  in  the  other  direction,  it 
is  necessary  to  move  the  pulley-guides  along  their 


FIG.  14. 

common  axis  until  the  condition  necessary  for  main- 
taining the  belt  in  position  is  fulfilled  for  this  particu- 
lar case.  It  must  be  remembered  that  the  two  pul- 
ley-guides rotate  in  contrary  directions,  and  therefore 
cannot  be  fixed  to  the  arbor  upon  which  they  run. 

From  the  arrangement  shown  in  Fig.  12,  that  of 
Fig.  14  may  be  devised  ;  this  disposition  corresponds 
to  the  case  in  which  there  is  a  very  slight  angle  between 
the  arbors,  and  the  pulley-guides  rotate  in  the  same 
direction. 


BELTS  AND  PULLEYS. 


The  disposition  represented  in  Fig.  15  is  still  more 
simple,  and  may  be  used  for  a 
greater  angle  between  the  axes — 
as  great  as  25°. 

FIG.  1 6.  Half-crossed  belt  ivith 
pulley-guide. — In  this  case  the  re- 
lative positions  of  the  pulleys  of 
transmission  are  such  that  the  dis- 
position represented  in  Fig.  9 
could  be  used,  except  that  the 
separation  of  the  pulleys  is  too 
slight,  and  the  belt  would  there- 
fore tend  to  run  off.  To  deter- 
mine the  arrangement  of  the  belt, 
we  begin  by  giving  to  the  part  55 
the  direction  of  the  line  of  intersection  of  the  middle 
planes  of  the  pulleys  A  and  B ;  then  from  the  point  c, 
chosen  arbitrarily  upon  the  line  55,  we  draw,  to  the 
circumferences  of  the  pulleys,  the  tangent  lines  ca  and 


FIG.  15. 


FIG 


cb.  The  plane  of  these  tangents  determines  the  middle 
plane  of  the  pulley-guide  C,  to  which  the  lines  are  also 
tangents.  Rotation  may  take  place  equally  well  in 
either  direction.  Because  of  the  cramped  position  of 


TRANSMISSIONS  BY  BELTS   WITH  GUIDES.      3/ 

the  pulleys  and  the  consequent  difficulty  in  placing 
the  arbor  of  the  pulley-guide  in  proper  position,  this 
arrangement  is  very  rarely  seen  in  practice. 

FIG.  17.  Another  disposition  for  transmission  by  half- 
crossed  belt  with  pulley-guide. — In  this  figure  the  pulleys 
of  transmission  are  so  placed  that  the  line  of  intersec- 
tion 55  of  their  middle  planes  is  the  common  tangent 
to  the  circles  contained  in  the  planes,  and  the  middle 
plane  of  the  pulley-guide  C  coincides  with  that  of  the 


FIG.  ,7. 

pulley  of  transmission  A.  The  portion  of  belt  which 
leaves  the  pulley  A  is  inclined  (as  shown  in  the  figure) 
as  in  the  crossed  belt  in  order  that  it  may  properly  roll 
upon  the  pulley  B,  while  the  portion  which  leaves  the 
pulley  .#  is  guided  by  the  pulley-guide  C.  The  pulley- 
guide  is  in  contact  with  the  line  of  intersection  55,  and 
with  a  tangent  to  the  circle  A  drawn  from  an  arbitrary 
point  upon  the  line  55.  In  this  disposition  the  direc- 
tion of  rotation  must  be  as  indicated  in  the  figure, 


3°  BELTS  AND  PULLEYS. 

This  mode  of  transmission  is  very  convenient  when  we 
wish  to  drive  a  series  of  vertical  arbors  from  one  hori- 
zontal shaft ;  it  also  finds  frequent  employment  in  mills 
for  grinding  various  materials,  and  when  the  separa- 
tion of  the  pulleys  of  transmission  is  necessarily  slight. 
FlG.  1 8.  Half-crossed  belt  with  movable  pulley-guide. 
— In  this  disposition,  which  is  used  for  a  greater  separa- 
tion of  the  pulleys  of  transmission  than  in  that  of  Fig. 
17,  we  may,  by  moving  the  pulley-guide  from  the  posi- 


FlG.    I 


tion  C  to  the  position  CQ  (shown  by  the  dotted  lines), 
cause  the  belt  to  pass  from  the  fixed  pulley  B  to  the 
idle  pulley  B^.  in  a  similar  manner,  the  pulley-guide 
may  be  used  for  running  the  belt  off  the  pulleys  en- 
tirely. The  position  CQ  should  be  so  chosen  that  the 
tensions  upon  the  belt  for  the  two  positions  will  be 
the  same  or  slightly  less  for  CQ  than  for  C. 

General  case  of  crossed  arbors. — When  the  pulleys  of 
transmission  cannot  be  so  placed  that  the  line  of  inter- 


7'RANSMISSIONS  BY  BELTS   WITH  GUIDES.      39 

section  of  their  middle  planes  is  a  common  tangent  to 
the  circles  contained  in  the  planes,  it  becomes  necessary 
to  make  use  of  two  pulley-guides.  Fig.  19  represents 
an  arrangement  which  may  be  adopted  in  such  cases, 
and  which  may  be  regarded  as  the  general  solution  of 
the  problem  of  transmission  by  belts  with  pulley- 


FIG.  19. 


FIG.  20. 


guides.  Fig.  20  represents  a  special  application  for 
the  case  in  which  the  line  of  intersection  5S  of  the 
middle  planes  passes  through  the  centre  of  the  middle 
circle  of  one  of  the  pulleys  of  transmission  ;  in  this 
figure  the  axis  of  the  pulley  B  is  supposed  to  be  situ- 
ated in  a  plane  parallel  to  the  pulley  A.  After  having 
obtained  the  line  of  intersection  55,  we  choose  upon 


40  BELTS  AND  PULLEYS. 

it  two  arbitrary  points  c  and  £,  through  which  we  draw, 
to  the  middle  circles  of  the  pulleys  of  transmission,  the 
tangent  lines  ca,  cb,  £,#„  and  cjb^.  The  planes  cab  and 
c\aJ)\  which  are  thus  determined  are  those  of  the  two 
pulley-guides,  which  should  be  placed  respectively  in 
contact  with  the  above-named  tangent  lines.  With 


Si 


FIG.  21. 

this  disposition,  rotation  may  take  place  equally  well 
in  either  direction. 

The  mode  of  transmission  represented  in  Fig.  19 
may  be  simplified  by  giving  to  the  axes  of  the  two 
pulley-guides  a  common  direction  mm  parallel  to  the 
two  pulleys  of  transmission  (Fig.  21).  In  this  figure  SS 
represents  the  intersection  of  the  middle  planes  of  the 


TRANSMISSIONS  BY  BELTS    WITH   GUIDES.      4! 

two  pulleys  of  transmission,  ac  and  &1c1  the  intersec- 
tions of  planes  perpendicular  to  55  with  the  middle 
planes  of  the  pulleys  of  transmission  A  and  B  respec- 
tively. In  the  perpendicular  planes,  tangentially  to  the 
right  lines  ac  and  bj^  we  place  the  two  pulley-guides 
C  and  Cr  The  arrows  indicate  the  directions  of  rota- 
tion ;  to  obtain  a  movement  of  the  belt  in  a  direction 
contrary  to  the  one  indicated,  it  is  necessary  to  give  to 


FIG.  22. 


the  pulley-guides  C  and  Cl  the  positions  indicated  at 
C  and  C\  by  the  dotted  lines. 

It  may  be  remarked  here  that  the  belt,  instead  of 
passing  from  c  to  a  and  from  cl  to  #,,  may  be  made  to 
pass  from  c  to  al  and  from  cl  to  a,  which  causes  a 
change  in  the  direction  of  rotation.  The  pulley-guides, 
instead  of  being  horizontal,  as  in  the  figure,  may  be 
placed  vertically — that  is,  respectively  in  the  planes  of 
the  pulleys  of  transmission  A  and  B ;  in  this  case,  how- 


BELTS  AND  PULLEYS. 


ever,  it  becomes  necessary  to  take  account  of  the  angle 
of  deviation  (see  §  4). 

When  the  pulleys  of  transmission  can  be  so  placed 
that  the  intersection  55  of  their  middle  planes  is  tan- 
gent to  one  of  the  pulleys,  and  the  distance  between 
the  parallel  planes  containing  the  axes  of  the  pulleys 
A  and  B  is  sufficient,  we  may  substitute,  for  the  dis- 
position shown  in  Fig.  20,  the  one  represented  in  Fig. 


FIG.  23. 

22.  This  arrangement  is  often  seen  in  practice ;  the 
axes  of  the  pulley-guides  are  parallel  to  that  of  the 
pulley  of  transmission  A.  The  middle  planes  of  the 
pulleys  A  and  B  may  make  any  desired  angle  with 
each  other. 

If  the  distance  AC  is  great  compared  with  the  width 
of  the  belt,  the  pulley-guides,  instead  of  being  the  one 
above  the  other,  may  be  placed  upon  the  same  axis,  as 
shown  in  Fig.  23.  If  the  distance  between  B  and  C  is 
sufficiently  great,  the  arbor  B  may  be  provided  with 
two  pulleys,  one  fixed  and  the  other  idle. 


TRANSMISSIONS  BY  BELTS   WITH  GUIDES.      43 

When,  on  account  of  lack  of  space,  it  is  impossible  to 
make  use  of  one  of  the  dispositions  which  we  have  de- 
scribed above,  we  ought  to  seek  at  least  to  place  the 
axes  of  the  pulley-guides  in  the  middle  plane  of  one  of 
the  principal  pulleys  and  the  pulley-guides  themselves 
parallel  to  each  other,  as,  for  example,  in  Fig.  24.  In 
this  case  we  first  draw  the  tangent  line  ab\  then  in  a 
plane  drawn  through  this  line  normally  to  the  plane  of 


FIG.  25. 

the  figure  we  place  the  pulley-guide  C  in  such  a  man- 
ner that  it  is  tangent  at  the  point  a  to  the  line  of  in- 
tersection of  the  middle  planes  of  the  pulleys  A  and  C. 
Through  the  point  a^  we  then  draw  the  line  a^  paral- 
lel to  ab,  and  in  a  plane  drawn  through  this  line  parallel 
to  the  plane  of  the  pulley-guide  C,  we  place  the  second 
pulley-guide  tangent  to  the  intersection  of  the  middle 
planes  of  the  pulleys  ^4  and  Cl  and  to  the  middle  plane 
of  the  pulley^.  In  this  manner  the  axes  mm  and  injn^^ 


44  BELTS  AND  PULLEYS. 

of  the  pulley-guides  are  found  parallel  to  each  other, 
and  also  situated  in  a  plane  parallel  to  that  of  the 
pulley  B. 

By  making  the  belt  of  Fig.  23  pass  over  a  fourth 
pulley  we  may  obtain  an  arrangement  by  which  we 
may  drive  two  pulleys  B  and  C  by  means  of  a  single 
driving-pulley  A. 

Fig.  25  represents  a  disposition  of  this  kind  much 
used  in  spinning-mills.  The  arbors  B  and  Care  in  dif- 
ferent stories  of  the  building,  and  each  bears  two  pul- 
leys, one  fixed  and  the  other  loose;  we  use,  in  this 


case,  the  permissible   deviation  of  the  belt    from  its 
exact  position  mentioned  in  §  3. 

Fig.  26  represents  another  mode  of  transmission  by 
belt,  in  which  the  two  parallel  arbors  B  and  C  are 
driven  by  a  single  pulley  A.  The  axes  of  these  arbors 
are  both  perpendicular  to  that  of  the  arbor  A  ;  the  first 
intersects  it,  while  the  second  crosses  it  without  inter- 
secting. In  the  machinery  of  spinning-mills  a  great 
number  of  transmissions  are  found  in  which  three,  four, 
or  even  a  greater  number  of  pulleys  are  driven  by 
means  of  a  single  driver.  It  may  be  remarked  here, 
that  in  all  cases  of  transmission  by  leather  belt  in 
which  pulley-guides  are  used  which  are  in  contact  with 


LENGTH  OF  BELTS. 


45 


the  upper  surface  of  the  belt,  it  is  advantageous  to 
place  the  belt  so  that  the  contact  of  the  pulleys  is 
always  upon  the  same  surface — the  flesh  or  wrinkled 
side. 

§  6.     Length  of  Belts. 

It  is  often  necessary  in  practice  to  calculate  the 
proper  length  of  a  belt  for  a  given  separation  of  the 
axes  of  the  pulleys  upon  which  the  belt  is  to  run  and 
for  known  pulley  radii  or  diameters.  Thus  when  we 
have  two  pulleys,  the  bearings  and  positions  of  which 


FIG.  27. 

are  already  fixed,  if  we  can  determine  the  proper  length 
for  the  belt,  we  can  save  time  and  prevent  waste  of 
belt  in  cutting  too  long  or  too  short. 

Open  Belt. — Let  us  denote  by  L  the  total  length  of 
the  required  belt;  by  Ll  the  distance  between  the 
centres  of  the  two  pulleys  upon  which  the  belt  is  to 
run  ;  by  R  the  radius  of  the  larger  pulley,  and  by  r  that 
of  the  smaller.  Let  Fig.  27  represent  the  pulleys  con- 


46  BELTS  AND  PULLEYS. 

nected  by  an  open  belt.  In  the  figure  the  lines  ob  and 
o'c  are  parallel,  because  each  is  perpendicular  to  the 
line  be ;  hence  the  angles  xob  and  yoc  are  equal.  Let 
us  denote  each  of  these  angles  by  (p.  It  is  evident 
from  the  figure  that  the  total  length  of  the  belt  must 
be 

L  =  2(bc  -f-  arc  ab  -f-  arc  cd\ 

Draw  the  line  ck  parallel  to  oo'  \  we  shall  have  ck  =  Lv 
because  ob  and  o'c  are  parallel.  In  the  triangle  bkc,  in 
which  the  angle  kbc  is  a  right  angle,  we  shall  have 


be  =      cl-  b&         or         be  =      Ll  --  6Jf- 

But  ok  =  o'c  —  r  and  bk  =  ob  —  ok  =  R  —  r\  hence 


be  =    VL*  -(R-  r)\ 

The  arc  ab  is  equal  to  the  arc  ax  -f-  the  arc  xb\  arc 

27tR  n  Rep 

ax  —  -    -  —  i.57£,  and  arc  4ri  ±=  -      -  = 

4  ibo 

Therefore 

arc  ab  —  i.tfR  +  0.0175^  =  (1.57  +  0. 


Also  the  arc  a/  is  equal  to  the  arc  dy  —  the  arc  yc\ 


2nr 
arc  ay  =  --  =  \.$jr  and  arc  yc  =  —  ^—  =  0.017  $rq>. 

Hence  we  shall  have 

arc  cd  =  i.$/r  —  0.017  $r<p  =  (1.57  —  0.0175^. 


//  YXS-* 

LENGTH   OF  BELTS.  47 

Add  together  these  values  of  be,  arc  ab,  and  arc  cd,  and 
we  shall  have  for  the  total  length  of  the  belt 


7-  o.oi75<p)r]  (24) 

In  the  right-angled  triangle  kbc  we  have,  from  trigo- 

bk      R  —  r 

nometry,  sin  angle  bck  —  i~  ~  —  7  —  •     But  s^nce  the 

sides  of  the  triangles  kbc  and  xbo  are  respectively  per- 
pendicular to  each  other,  the  triangles  are  similar,  and 
the  angle  bck  •=.  <p.  Hence  we  shall  have 

sin  v  ±=  —  rr-.*    .....     (25) 
f*i 

Crossed  Belt.  —  In  crossed  belts  the  lengths  differ  con- 
siderably from  those  of  open  belts  under  the  same  cir- 

*  In  open  belts  the  angle  cp  is  generally  quite  small,  and  we  may 
without  serious  error  take  the  sine  of  th*  angle  equal  to  the  angle  it- 
self expressed  in  circular  measure.  Thus  we  shall  have,  from  for- 

R-r  R-r 

mula  (25),  <pc  =  sin  cp  =  —  -  —  =  o.oi75<p,  or  cp  =  —     —  —  ,  cp    repre- 


senting  the  angle  in  circular  measure.     This  value  substituted  in  for- 
mula (24)  gives  for  the  total  length  of  the  belt 


^2Tj  =  2I     «T?~(R  -  r)*+i.s7(X+ 


Representing  R  -f-  r  by  2  and  R  —r  by  A,  the  above  expression  be- 
comes 


~         .     .     .     (25A) 
*•»/ 

This  formula  is  simpler  than  but  not  so  accurate  as  formula  (24). 


48 


BELTS  AND  PULLEYS. 


cumstances  of  separation  of  pulleys  and  pulley-radii. 
Let  Fig.  28  represent  two  pulleys  connected  by  a 
crossed  belt.  As  in  open  belts,  we  shall  have 

L  —  2(bc  +  arc  ab  -f-  arc  cd\ 

Draw  the  line  ck  parallel  to  oof  and  produce  bo  as  far 
as  its  intersection  with  ck.  As  before,  we  shall  have 
ok  =  o'c  =  r.  Hence  bk  =  R  -f-  r  and  ck  =  L^ 


y 


FIG.  28. 


In  the  right-angled  triangle  kbc  we  shall  have 

be  = 


The  triangles  ^^r  and  /^fc  are  similar,  and  the  angle 
bck  =:  99.     We  shall  therefore  have 


sin  angle  bck  =  T-, 


or 


sm  <p  =  — 


(26) 


LENGTH  OF  BELTS.  49 

From  the  figure  we  have  arc  ab  —  arc  ax  -\-  arc  xb  = 

27tR 

— \-  o.oi75/t<p.     Also  arc  cd  —  arc  yd  +-  arc  yc  = 

2nr 

— -+o.oi7Sr<p. 

Hence,  by  adding  together  these  values  of  be,  arc  ah, 
and  arc  cd,  we  shall  obtain  for  the  total  length 


L  =  2[  VLi2  —  (y?  +  r)'2+  1.57^  +  0.0175^^)+  i.57r+o.oi75r<p], 
or 

]•    (27) 


Example  I.  —  Suppose  we  have  two  pulleys  having 
radii  of  R  =  20"  and  r  =  10",  and  a  distance  between 
the  axes  of  L,  =  10'  =  120".  Required  the  length  for 
an  open  belt  which  will  properly  connect  the  two  pul- 
leys. From  formula  (25)  we  shall  have 

20  —  10  47° 

sm  <p  =       I2Q       =  0.0833,  cp  =  4-^-. 

Formula  (24)  therefore  becomes,  on  substituting  the 
,  above  data, 


=  2     V'l^2  —  (20  —  io)2+  f  1. 


—  (20  —  io)2+    1.57  +  o  0175  X  4    o 

-00175x4^1' 

or 

L  =  2(119.  58  +  33-07  +  14-86)  =  335.02"  =  27'  n". 


^' 


If  we  wish  to  use  formula  (2  5  A)  instead  of  formula 
(24),  we  proceed  as  follows  :  2  =  20  +  10  —  30,  A  = 


50  BELTS  AND  PULLEYS. 

20  —  10  =  10.     Hence,  from  the  formula,  we  shall  have 


=  2(1  19.58  +  47-10  +  0.833)  =  335-03". 

Thus  the  difference  in  the  results  from  formulas  (24) 
and  (2  5  A)  is  in  this  case  practically  o. 

Example  2.  —  Taking  the  data  of  Example  I,  it  is 
required  to  calculate  the  proper  length  for  a  crossed 
belt  which  runs  on  the  above  pulleys.  From  formula 
(26)  we  shall  have 

20  +  10  29° 

sm?  =  ~i^r~  =a2s»     or     ?>  =  I4  -<£•==  14-5  • 

Formula  (27)  therefore  gives  for  the  proper  length  of 
the  belt 

L  —  2[|/I20*  —  (20  +  I0)a  +  I.57(20  +  10)  +  0.0175 
X   145  (20+  10)]  =  2(ll 


or  L  =  341.80"  =  28J  feet. 

In  transmissions  by  belts  with  pulley-guides,  and  in 
all  cases  where  the  intricate  arrangement  of  the  belt 
renders  arithmetical  calculation  long  and  tedious,  the 
proper  length  of  the  belt  may  be  determined  more 
easily  and  with  sufficient  exactness  graphically,  by 
measurement  with  the  rule.  To  illustrate:  Suppose  we 
have  an  arrangement  of  pulleys  such  as  is  represented 
in  Fig.  12,  which  figure  is  a  sketch  (containing  two 
projections)  of  the  transmission,  drawn  to  a  scale  of  ^. 


SPEED-CONES.  5 1 

The  separation  of  the  pulleys  A  and  B  is  5  feet  =  60", 
and  the  diameters  respectively  2 \"  and  13".  From  the 
figure  it  is  evident  that  the  total  length  of  the  belt  is 
L  =  arc  xy+yD  +  Dx'  +  arc  x'y'  +  y'C  +  Cx.  By 
measuring  with  the  compasses  the  above  arcs,  we  find 
xy  =  34i"  and  x'y'  =  20".  The  line  yD  in  the  left- 
hand  projection  is  given  in  its  true  length  by  the  line 
ND  in  the  right-hand  figure  ;  hence,  by  measuring 
ND,  we  obtain  for  the  true  length  yD  —  30".  The 
distance  Dx'  is  given  in  its  true  length  in  the  left-hand 
figure,  and  therefore,  by  direct  measurement,  we  ob- 
tain Dx'  =  34/x.  In  a  similar  manner  we  obtain  by 
measuring  KC  the  true  length  y' C  =  36",  and  by  di- 
rect measurement,  Cx  =  26".  We  have  consequently 
L  =  34*  +  30  +  34  +  20  +  36  +  26  =  1 80*"  =15'  £". 
In  a  similar  manner  in  Fig.  18,  by  measuring  the  arcs 
xy  and  x'y'  ^  and  the  length,  NX,  Ky ',  and  yz,  we  may 
obtain  the  length  necessary  for  a  belt  which  will 
properly  run  on  the  pulleys  represented  in  the  figure. 

§  7.     Speed-cones. 

The  contrivance  known  to  mechanics  as  "  speed- 
cones"  consists  of  two  stepped  pulleys  arranged  as 
shown  in  Fig.  29.  The  object  of  speed-cones  is  to  ob- 
tain different  speeds  for  the  driven  arbor  from  the  con- 
stant speed  of  the  driving-shaft.  To  illustrate  :  Suppose 
in  Fig.  29  we  assume  between  the  radii  of  the  pulleys 
the  relations  R  =  37-,  R'  —  r' ,  and  R'  =  ^r" .  We  have 
seen  from  formula  (6)  that  the  ratio  of  the  revolutions 
of  two  pulleys  which  are  connected  by  one  and  the 
same  belt  is  equal  to  the  inverse  ratio  of  the  pulley- 


52  BELTS  AND  PULLEYS. 

radii.  Hence,  if  we  assume  that  the  driving-shaft  xy 
makes  100  revolutions  per  minute  (N=  100)  when  the 
belt  is  on  the  pulleys  R  and  r,  we  shall  have  for  the 
revolutions  of  r  (and  consequently  of  the  shaft  x'yr) 

r> 

n  —  N—  —  100  X  3  =  300.     When  the  belt  is  on  the 

pulleys  R  and  r'  we  shall  have  for  the  revolutions  of 

R' 

T*  (and  consequently  of  the  shaft  x'yf)  n'  =  N  —f  =  100 

X  I  =  IOO.     Similarly,  when  the  belt  is  on  the  pulleys 


X~ 


L, 


FIG    29. 


R"  and  r"  we  have  for  the  revolutions  of  R"  and  the 

r" 
shaft  x'y'  N"  =  N  ~^r  =  100  X  \  =  33f      Such  differ- 

ent speeds  for  the  driven  arbor  are  necessary  in  many 
machine-tools,  as  the  lathe,  drill,  etc.,  because  the 
speed  of  the  mandrel  and  spindle  must  vary  with  the 


SPEED-CONES.  S3 

material  to  be  worked  and  with  character  of  the  work 
to  be  done. 

Open  Belt.  —  From  formula  (25A)  we   have   for  the 
lenth  of  the  belt 


1.572  +- 


in  which  2  =  R  -f-  r  and  A  =  R  —  r  (see  Fig.  29). 
Since  now  the  length  of  the  belt  must  be  the  same  for 
each  pair  of  pulleys  in  the  cone,  we  shall  have 


2(  ^Z7= 


in  which  2'  =  R  +  r'  and  A'  =  Rr  -  rf. 

By  means  of  the  binomial  formula  we  may  extract 
the  square  roots  of  the  quantities  under  the  radical 
signs  as  follows : 


and 


„  J'2         J'4         J/B 

But  since  Zt  is  usually  very  large  compared  with  //, 

z/4  /I6 

^y-j  and    ,-j  6  are  very  small  quantities,  and  may  with- 


54  BELTS  AND  PULLEYS. 

out   serious   error   be    neglected.      Similarly,  we  may 

A"  Ar* 

neglect  the  quantities  ^r-3  and     „  5.     Hence  we  shall 

have 


which  reduces  to 


If  we  represent  by  JVthe  constant  number  of  revolu- 
tions per  minute  of  the  driving-shaft  (corresponding  to 
R),  and  by  n  the  number  of  revolutions  per  minute  of 
the  driven  shaft  when  the  belt  is  on  the  pulley  r,  we 
shall  have,  from  formula  (6), 

R         n  n 

_  =  -,        or        **?r^ 

R         n'         n'  ,  ri 

Also          -^  =  w,  =  ^,         or         R=*^ 

in  which  ;^r  represents  the  revolutions  per  minute  of 
the  driven  shaft  when  the  belt  is  on  the  pulleys^"  and 
r'  ,  and  N'  the  revolutions  per  minute  of  the  driving- 
shaft.  which  being  constant  is  equal  to  N.  Hence  we 
shall  have 


SPEED-CONES.  55 

which  substituted  in  formula  (28)  gives 


We  shall  also  have  (as  above  for  the  quantity  2') 
4>  =  R'  -  r'  =  rf  -      -  /, 


or  4  =  r'(-jf  --  ij (30) 

Example  I. — Suppose  we  have  two  shafts,  the  dis- 
tance between  which  is  Ll  =  100":  the  revolutions 
per  minute  of  the  driving-shaft  is  N=  100,  and  we 
wish  to  construct  a  pair  of  speed-cones  such  that  the 
revolutions  per  minute  of  the  driven  shaft  correspond- 
ing to  the  pulleys  r,  r ',  r" ,  and  R"  shall  be  ;/  ==  300, 
ri  =  200,  n"  —  100,  and  N'"  —  50.  From  formula  (6) 
we  shall  have 

£         n         300 

-\  T      1 1  O1*  1\.      3  ?*. 

r        N        100 

We  may  choose  any  convenient  value  for  r,  and  find 
from  the  above  expression  the  corresponding  value  of 
R.  Suppose  we  take  r  =  4" ;  hence  R  =  ^r  =  3  X  4 
—  12".  Then  2  =  12  -f  4  =  16  and  J  =  12—4 
=  8.  From  formula  (30)  we  shall  have 


A'  =  r'(-- 
Vioo 


5  6  BELTS  AND  PULLEYS. 

and  formula  (29)  becomes 

.(200          \  64  —  r'*  ,-64—  r'\ 

r'[  ---  \-  i    =  16  +  —  —  ,pr3r'=  16  +  - 

\ioo          /  r  3.  14x100'  314 

From  this  by  reducing  we  shall  have 

r'2  +  942?-'  =  5024  +  64. 

Adding  (  --  j   =  47i2  to  each  side  of  this  equation  gives 

r"  +  942r'+  221841  =  5024  +  64-)-  221841  =  226929. 
Extracting  the  square  root  of  this  expression  gives 

r1  +  471  =    ^226929. 
From  this  r7  =    1/226929  —  471, 

or  '  r'  =  476.38  —  471  =  5.38". 


Then       -pr  rrr  ^r   =  -,  Or  ^  =  2/  =    10.76^. 

In  the  same  manner  for  the  pulleys  T?"  and  r"  we 
shall  have  from  formula  (30) 


and  formula  (29)  becomes 


—  2r"  =  16  -1 =  16.204, 

100    '     /  314 


or 


Also         ^  =  —  3,         or         /?//  =  r//  =  8.i02//. 
r         100 


SPEED-CONES. 
For  the  pulleys  r"  and  Rf"  we  shall  have 


57 


Hence  formula  (29)  gives 

y>"  =  16  +  ^^—  ,    942^  -  5024  +  64  -  r"'2. 
Hence  r///2  +  942^"  =  5088. 

As  before,  adding  (~")   to  each  side,  and    extracting 
roots,  we  shall  have 


r!"  =  1/5088  +  221841  —  471  =  5.38". 
Then 

*£=.£/==!°°  =  2f        or        jr=2r"'=io76". 


Crossed  Belt. — The  calculation   of  the  radii  of   the 
speed-cone  pulleys  becomes  very  much  simpler  when 


58  BELTS  AND  PULLEYS. 

crossed  belts  are  used.  If,  in  Fig.  30,  we  assume  the 
relations  2  =  R  +  r  =  Rf  +  r'  =  R"  +  r"  ,  etc.,  we 
shall  have  for  the  corresponding  angles,  cp,  g>',  g>",  etc.; 

R  +  r       2     .       ,      R+r'       2     . 
sin  cp  =  —  j  --  =  y-,  sin  cp  =  --  j  ---  =  —   sin  <p"  = 

**i        *+\  **\        4*1 

7?//    |    f  r/         "V 
—  j—  -  =  y  ,  and  consequently  9?  =  cp'  •=.  q>f/,  etc. 

£r,  L^ 

The  conditions  that  the  length  of  the  belt  must  be  the 
same  for  each  pair  of  pulleys,  and  that  the  belt  must 
bear  the  same  tension  for  each  pair  of  pulleys,  will 
therefore  be  fulfilled  if  we  take  the  sums  of  the  radii 
of  each  pair  of  pulleys  equal  to  each  other.  Or,  which 
is  the  same  thing,  we  shall  have 

R'  =  ^-rr  .......     (31) 

Letting  R  -f-  rf  =  2',  we  shall  have  from  above 

2  =2'. 
From  formula  (6)  we  may  write 

R        n  n 

-  =   Tr  or  K  —  r^~r, 

r        N  N 

R       nf       nf  ,ri 

~'='=-  =r* 


Hence  2  = 


=  r(~  +  i), 


SPEED-CONES.  59 

'+•=    +  •• 


Example  2.  —  Taking  the  data  of  Example  i,  it  is 
required  to  calculate  the  radii  of  the  speed-cone  pul- 
leys for  crossed  belt.  We  obtain,  as  in  Example  i, 
R  —  12",  r  —  4",  and  2  =  16".  From  formula  (32) 
we  shall  then  have 

OO\  4 

=  4  x  3  =  5>33  ' 


Formula  (31)  then  gives 

R'  =  16  -  5.33  =  10.67". 
For  the  third  pair  of  pulleys  formula  (32)  gives 

_,,  _  Jn  +  N\  _     (300+  ioo\  _  _  „„ 

rU"  +  JW      4\ioo  +  ioo/ 

and  from  formula  (31)  we  shall  have 

For  the  fourth  pair  of  pulleys  from  formula  (32)  we 
shall  have 


Formula  (31)  then  gives 

rx//  =  ^  -  R'"  =  16  -  10.67  =  5.33^. 


60  BELTS  AND  PULLEYS. 

Suppose  now  that  we  wish  to  add  to  the  speed-cones 
another  pair  of  pulleys  (Riv  and  riv)  having  such  radii 
that  the  number  of  revolutions  per  minute  of  the 
driven  shaft,  when  they  are  in  use,  shall  be  JViv  =  33^. 

We  shall  have  from  formula  (32) 

300  +  ioo\ 

=4x3-  12", 


and  from  formula  (31) 


We  have  now  two  speed-cones,  which  are  made  up 
of  pulleys  as  follows  : 

First  Cone.  Second  Cone. 

R    =  1.2"  r      =    4" 

R'   =  10.67"  r'      =     5.33" 

R"  =  8"  r"     =    8/r 

r'"  =  5.33^  R"'  =  10.67" 

r™  =  4"  ^iv   =  I27/ 

A  glance  at  this  table  will  show  that  the  two  cones 
are  similar  and  equal,  but  so  placed  on  their  shafts 
that  they  taper  in  opposite  directions.  We  may  there- 
fore write  the  following: 

Rule  for  Speed-cones,  Crossed  Belt.  —  Use  two  equal 
and  similar  stepped  cones  tapering  in  opposite  direc- 
tions. 

Mr.  C.  A.  Smith,  in  the  American  Machinist,  Feb- 
ruary 25,  1882,  gives  a  very  neat  graphical  method  for 
determining  the  radii  of  speed-cone  pulleys  for  open 
belt,  as  follows:  Lay  off  (Fig.  31)  AB  equal  to  the 


SPEED-CONES. 


Vc,, 


given  distance  between  the  two  shafts  (AB  -=.  Z,), 
drawn  to  any  convenient  scale.  Strike  the  circles  repre- 
senting the  pulleys  R  and  r  (the  radii  of  which  are  deter- 
mined, as  in  Examples  I  and  2  of  this  section,  from  the 

given  revolution-ratio  -^.j,  and  draw  the  portion  of  belt 


ab.     Lay  off   (from   the   smaller  pulley-centre)  BC  = 
AB  X  0.496  —  0.496^,  and    erect    the    perpendicular 

CD  =  -     —  >.     Then  from  D  as  a  centre  strike  the  cir- 
3.1416 

cle  x  tangent  to  ab.     Divide  AB  =  Ll  into  as  many 


FIG.  31. 


equal  parts  as  the  shaft  B  is  to  revolve,  less  one,  while 
the  shaft  A  makes  one  revolution,  when  the  belt  is  on 
the  required  pulleys  R'  and  r  '.  Lay  off,  from  the  cen- 
tre of  the  smaller  pulley,  BO  equal  to  one  of  these 

parts  (BO  =  L1  -r-  -^  —  i),  and  from  o  draw  the  line  oa' 


tangent  to  the  circle  x. 


The  circles  drawn  from  B  and 
A  as  centres  and  tangent  to  oa'  give  the  required 
radii  r'  and  R'.  When  we  wish  to  have  the  revolutions 


62 


BELTS  AND   PULLEYS. 


of  the  driven  shaft  B  less  than  those  of  the  driving- 
shaft  A,  or  when  the  smaller  pulley  is  to  be  on  the 
shaft  A,  we  lay  off  (for  r'"  and  R'f/)  the  distance  Ao'  = 

N 
Ll  -r-  T^777  —  I,  draw  o'b'  tangent  to  the  circle  x,  and 

the  circles  r"  and  R'"  give  the  required  radii. 

Crossed  belts  are  not  so  often  used  for  speed-cones 
as  open  belts,  and  the  speed-cones  for  the  former  are 
so  easily  calculated  from  formula  (32),  that  it  is  un- 
necessary to  give  graphical 
methods  for  determining 
the  radii. 

Continuous  Speed-cones. — 
Sometimes  (especially  in 
cotton  machinery  and  in 
machines  requiring  gradu- 
ally increasing  or  decreas- 
ing speeds  for  the  driven 
arbors)  continuous  speed- 
cones  are  used  instead  of 
the  stepped  speed-cones 
already  described.  It  may, 
however,  be  remarked  that 
in  ordinary  shop  machin- 
ery, such  as  lathes,  planers, 
drills,  etc.,  etc.,  continuous 
speed-cones  are  very  rarely 
seen. 

To  construct  a  pair  of  continuous  speed-cones  for 
open  belt  we  may  proceed  as  follows :  Having  given 
several  of  the  different  numbers  of  revolutions  re- 
quired of  the  driven  shaft  (for  example,  n  —  300,  #'=  — , 


FIG.  32. 


SPEED-CONES. 


rif  =  100,  N"'  —  — ,  Niv  =  50,  and  the  revolutions  of 
the  driving  shaft  being  'N=  100),  lay  off  (Fig.  32) 
ab  ==  a!bf  =  the  width  of  the  belt  -f-  the  proper  clear- 
ance X  the  number  of  changes  in  the  speed  of  the 
driven  shaft:  in  this  case  there  are  five  changes. 
Then  calculate,  from  formulas  (29)  and  (30),  the  radii 
R,  r,  R ',  rf ',  riv,  and  Klv,  corresponding  to  the  known 


FIG.  33. 


FIG. 


numbers  of  revolutions,  and  draw  the  pulleys  of  which 
R,  r,  etc.,  are  the  radii,  and  which  are  represented  by 
the  dotted  rectangles  in  the  figure.  Through  the  cen- 
tres of  the  step-widths  (x,  y,  z,  x ',  etc.)  draw  the  curves 
xy%i  x'y'z',  and  the  outlines  of  the  cones  are  complete. 
Rankine  gives  for  continuous  speed-cones  for  open 
belt  the  rule,  "•  Use  two  equal  and  similar  conoids  taper- 


64 


BELTS  AND  PULLEYS. 


ing  in  opposite  ways  and  bulging  'in  the  middle,  accord- 

r  _i_  r        lr  _  r 
-  -  2  +     t    ^  2 


in     to  the  formula  rQ  — 


in  which 


r0  is  the  radius  in  the  middle,  r,  and  r2  the  radii  of  the 
larger  and  smaller  ends  respectively,  and  c  the  distance 
between  the  centres  of  the  shafts.  Fig.  33  represents 
a  pair  of  continuous  speed-cones,  open  belt,  calculated 
from  this  rule,  taking  r^  =  10",  r2  =  4",  c  =  loo", 

-  =  7-057",  and  ab  =  a'V  =  14". 


To  construct  a  pair  of  continuous  speed-cones  for 


FIG.  35. 

crossed  belt,  calculate  from  formula  (32)  the  radii  R,  r, 
R",  r",  riv,  R[v  (Fig.  34),  and  connect  the  centres  of 
the  step-widths  by  the  curves  xyz,  x'y ' z' ',  in  the  same 
manner  as  in  Fig.  32.  Or  we  may  use  two  equal  and 
similar  cones  tapering  in  opposite  directions  (Fig.  35). 
An  example  will  best  explain  the  mode  of  calcula- 
tion for  a  pair  of  continuous  speed-cones  by  which  we 
wish  to  obtain  a  given  gradual  change  in  the  speed 
of  the  driven  arbor.  Suppose  our  driver  makes  100 
revolutions  per  minute,  and  that  we  wish,  by  slowly 


MATERIALS    USED  FOR  BELTING.  65 

sliding  the  belt  along  the  cones,  to  obtain  for  the 
driven  arbor  a  speed  varying  from  100  to  10  revolu- 
tions per  minute.  According  to  the  rapidity  with 
which  we  wish  the  changes  to  take  place  we  choose 
the  number  of  changes — let  us  say  in  this  instance 
10.  Of  these  changes,  the  number  of  revolutions  per 
minute  of  the  first  is  100.  With  the  9  remaining 
changes  we  must  therefore  gain  100  —  10  =  90  revolu- 
tions per  minute,  or  10  each.  The  revolutions  of  the 
changes  are  therefore  as  follows:  1st,  100 ;  2d,  90; 
3d,  80;  4th,  70;  5th,  60;  6th,  50;  7th,4o;  8th,  30;  gth, 
20  ;  loth,  10.  We  may  now  calculate  the  diameters  as 
for  stepped  cones,  and  by  drawing  curves  through  their 
face-centres  obtain  the  outlines  for  the  required  con- 
tinuous cones.* 


§  8.     Materials  used  for  Belting. 

• 
Belts  are  most  commonly  made  of  leather,  cut  into 

strips  of  the  required  width,  and  riveted  together  at 
their  ends  to  make  up  the  required  length.  Strips 
taken  from  the  back  part  of  the  hide,  and  oak  or  hem- 
lock tanned,  are  generally  considered  the  best,  although 
some  kinds  of  patent-tanned  leather  are  said  to  have 
greater  adhesive  power.  Cow's  hide  is  almost  invari- 
ably used  for  the  leather  of  belts ;  the  skins  of  horses, 
elephants,  and  other  animals  have,  however,  been  util- 

*  In  designing  continuous  speed-cones  it  is  always  best  to  make  the 
curves  as  gradual  in  taper  as  possible  for  the  given  changes,  in  order 
to  avoid  the  excessive  stretching  and  wear  of  the  belt  which  would 
otherwise  occur. 


66  BELTS  AND   PULLEYS. 

ized  for  this  purpose,  in  some  cases  with  very  good 
results.  For  very  heavy  work,  belts  made  of  two  or 
more  thicknesses  of  leather  are  used,  in  which  case  the 
strips  are  fastened  together  with  cement  or  rivets,  and 
the  joints  carefully  "  broken/'  In  order  to  gain  strength 
and  prevent  stretching,  leather  belts  are  sometimes 
edged  on  the  upper  side  with  narrow  strips  of  leather, 
which  are  riveted,  laced,  or  cemented  fast  to  the  belts. 
It  has  also  been  proposed  (and  to  our  knowledge  in 
one  case  at  least  tried)  to  strengthen  belts  by  riveting 
along  their  edges  thin  strips  of  brass,  steel,  or  other 
metals. 

Of  late  years  vulcanized- rubber  belts  have  been  very 
successfully  introduced  in  this  country.  They  are 
usually  made  continuous,  thus  avoiding  the  use  of 
rivets,  and  consist  of  one  or  more  layers  of  cotton-duck 
placed  between  layers  of  vulcanized  rubber,  the  rubber 
covering  the  edges  in  order  to  protect  the  seams  from 
injury.  Rubber  belts  are  now  made  in  widths  about 
the  same  as  leather;  they  weigh  nearly  the  same,  and 
are  said  to  be  equally  strong  and  pliable. 

The  intestines  of  sheep,  cats,  and  other  animals  have 
been  used  to  a  considerable  extent  for  belts  ;  they  are 
exceedingly  strong  and  tough,  and  can  be  obtained,  it 
is  said,  thirty,  or  forty  feet  in  length.  Gut  belts  are 
either  round,  to  run  in  grooved  pulleys,  or  woven  into 
flat  bands  for  use  on  ordinary  flat-faced  pulleys.  Raw- 
hide possesses,  it  is  claimed,  fifty  per  cent  more 
strength  than  tanned  leather ;  but  belts  of  this  material, 
unless  constantly  oiled,  soon  become  stiff  and  ungov- 
ernable, and  are  not  to  be  depended  upon  for  general 
purposes  of  transmission.  Belts  of  hemp,  flax,  canvas, 


MATERIALS   USED  FOR  BELTING.  6/ 

sheet-iron  and  steel,  and  several  combinations  of  leather 
and  metallic  wire,  have  been  proposed,  and  in  some 
cases  used  ;  but  these  at  present  offer  no  practical 
advantages  over  leather  and  vulcanized  rubber. 

For  all  practical  purposes,  then,  we  have  two  kinds 
of  belting — leather  and  rubber,  between  which  we  may 
offer  the  following  comparison :  Those  who  favor 
leather  belts  claim  that  they  are  in  the  main  stronger 
than  rubber,  and  that  they  will  wear  much  longer, 
especially  when  used  for  cross  or  half-cross  pulleys ; 
that  leather  belts  cease  to  stretch  after  once  or  twice 
shortened  and  relaced,  while  those  of  rubber  do  not  ; 
and  that  leather  will  bear  contact  with  oil  and  grease 
without  harm,  while  rubber  thus  exposed  will  soften, 
and  stretch  out  of  shape.  Wide  leather  belts  can  be 
cut  up  into  narrow  ones,  while  rubber  belts  cannot  be 
cut  without  injuring  the  finished  edges  ;  also,  leather 
can  be  more  easily  repaired  when  injured  than  rubber. 
On  the  other  hand,  rubber  belts  do  not  need  to  be 
riveted,  but  are  made  continuous ;  they  do  not  slip  so 
easily  on  the  pulley-faces  as  leather,  and  are  cheaper 
at  first  cost  for  the  same  sizes.  It  is  also  claimed  that 
'rubber  belts  endure  exposure  to  cold  and  wet  much 
better  than  leather,  retain  their  flexibility  better,  and 
do  not  lose  strength  so  rapidly  from  wear.  Leather 
and  vulcanized  belts  both  are  good.  Thousands  of 
each  perform  well  their  arduous  duties  all  over  the 
civilized  world.  Each  has  hundreds  of  admirers  and 
champions.  We  therefore  deem  it  best  to  express  no 
preference  on  our  own  part,  preferring  rather  to  have 
each  purchaser  choose  for  himself,  assuring  him  that 
either  good  leather  or  ijood  vulcanized  rubber  will  do 


68  BELTS  AND  PULLEYS. 

his  work  as  faithfully  and  well  as  any  reasonable  man 
should  desire. 

§  9.    Lacing  and  other  Modes  of  Fastening. 

Endless  belts,  of  whatever  material  they  are  made, 
when  subjected  to  a  considerable  strain  for  any  length 
of  time  become  lengthened  or  stretched.  As  a  result 
of  this  lengthening,  the  belts  hang  loosely  upon  their 
pulleys,  and  consequently  slip  and  slide.  It  is  there- 
fore necessary  to  have  some  ready  means  of  shortening 
belts  to  their  proper  lengths,  and  thus  make  them  again 
fit  tight  upon  the  pulley-faces.  This  is  very  generally 
done  by  leaving  the  belt  with  two  ends  (i.e.,  not  end- 
less), and  then  lacing  together  the  free  ends  with  leather 
thongs  or  cords.  When  a  laced  belt  becomes  stretched, 
it  is  unlaced,  cut  off  to  the  proper  length,  and  laced  up 
again,  new  holes  having  been  punched  at  the  cut  end."* 

Lacing-thongs  are  commonly  made  of  leather  or 
good  clean  rawhide,  softened  and  stretched  somewhat 
to  render  it  firm  and  even  ;  they  vary  in  width  from 
one  quarter  to  three  quarters  of  an  inch,  and  in  thick- 
ness from  one  sixty-fourth  to  nearly  one  eighth  of  an 
inch,  according  to  the  width.  We  may  say  very  simply, 
in  lacing  belts,  punch  the  holes  just  large  enough  to 
easily  admit  the  lacing-thong  f  inch  to  I  inch  from  the 
ends  of  the  belt  (no  more  material  than  is  necessary 

*  Sometimes  belts  of  considerable  length  are  shortened  to  take  up 
the  stretch  by  simply  running  off  one  pulley  and  twisting  the  belts 
until  the  proper  lengths  are  obtained.  This  practice  is,  however,  a 
very  bad  one,  because  the  twists  cause  the  belts  to  become  cracked 
and  to  wear  out  rapidly,  and  should  never  be  indulged  in  except  in 
cases  of  immediate  necessity. 


LACING  AND   OTHER  MODES  OF  FASTENING*     69 


should  be  cut  out,  because  this  tends  to  weaken  the 
belt) ;  use  for  small  belts  a  -J-inch  thong  ;  for  belts  from 
4  inches  to  8  inches  wide,  a  f-inch  thong  ;  for  belts  from 
8  inches  to  15  inches  wide,  a  J-inch  thong;  and  for  belts 
over  1 5  inches  in  width,  a  f-inch  thong.  The  first  requi- 
site in  lacing  together  the  free  belt-ends  is  to  have  the 
ends  square — that  is,  at  right  angles  with  the  sides  of 
the  belt ;  if  the  ends  are  not  square  the  belt  will  not  lie 
straight  on  the  pulleys,  and  will  tend,  consequently,  to 


FIG.  36.  FIG.  37. 

run  off  the  pulleys,  and  otherwise  interfere  with  the 
proper  motion  of  the  machine. 

The  simplest  mode  of  lacing  belts,  which  is  repre- 
sented in  Fig.  36,  consists  in  starting  at  one  side,  and 
lacing  over  and  over  through  all  the  holes  until  the 
other  side  of  the  belt  is  reached.  This  does  well 
enough  for  small  belts  not  to  be  subjected  to  any 
severe  strain,  although  even  they  will  do  more  satis- 


BELTS  AND  PULLEYS. 


factory  work  if  laced  differently ;  but  for  larger  belts 
better  and  safer  methods  must  be  used. 

Fig.  37  shows  a  style  of  lacing  quite  common  among 
machinists,  and  which  combines  quickness  of  operation 
with  strength  about  as  well  as  any  of  the  simpler 
methods.  Begin  at  the  side  a  in  the  figure,  and  lace 
with  both  ends  of  the  thong,  as  shown,  fastening  the 
ends  at  b  in  a  knot  or  other  convenient  manner. 

A  still  better  lacing  is  represented  in  Fig.  38.  The 
thong  is  here  crossed  on  one  side  of  the  belt  only— 
the  upper  side,  and  care  should  be  taken  not  to  cross 
unevenly  the  double  parts  on  the  pulley-side. 

In  heavy-driving  belts,  and  in  all  belts  where  the 
strain  is  severe,  double  rows  of  holes  should  be  punched, 
and  the  joining  thus  rendered 
doubly  secure  against  breakage. 
Messrs.  J.  B.  Hoyt  &  Co.,  manu- 
facturers of  leather  belting,  New 
York,  inform  me  that  all  their 
belts  are  laced  according  to  the 
double  method  represented  in 
Fig.  39,  in  which  a  is  the  side 
to  be  placed  next  the  pulley. 
This  lacing  has  the  advantage 
that  all  its  parts  on  the  outside 
of  the  belt  are  parallel  to  the  di- 
rection of  motion,  and  the  ten- 
dency is  therefore  to  keep  the  ends  of  the  belt  at  all 
times  in  their  proper  positions.  The  above-mentioned 
gentlemen,  after  many  years  of  experience  with  leather 
belting,  have  come  to  believe  this  method  the  best  in 
ordinary  use. 


FIG.  38. 


LACING  AND   OTHER   MODES   OF  FASTENING-      7 1 


An  excellent  style  of  lacing  for  large  belts  is  given 
by  Mr.  John  W.  Cooper  in  his  "  Use  of  Belting/'  which 


FIG.  39. 

we  represent  in  Fig.  40.  Begin  with  one  end  of  the 
lacing-thong  at  the  point  a,  and  lace  successively 
through  the  holes  i,  2,  3,  4,  5,  and 
so  on,  all  around  the  rows  of  holes 
until  the  point  a  is  again  reached, 
where  the  thong  is  fastened  off  as 
,shown  in  the  figure.  Although  in 
this  case  the  parts  of  the  thong  are 
not  parallel  to  the  direction  of  mo- 
tion, yet  they  are  so  slanted — on 
the  pulley-side  in  one  direction  and 
on  the  outside  equally  in  the  other 
— that  the  result  is  practically  the 
same,  and  the  lacing  is,  beyond 
doubt,  one  of  the  best  in  existence. 
Several  kinds  of  metallic  belt- 
hooks  or  fasteners  have  been  from 


FIG.  40. 

time  to  time  con- 


BELTS  AND  PULLEYS. 


trived  and  introduced — never,  however,  to  our  knowl- 
edge, with  any  great  degree  of  success.  For  small  belts 
the  best  of  these  hooks  do  well  enough,  and  lessen  the 
work  of  relacing  and  shortening ;  but  large  driving- 
belts,  and  those  used  to  transmit  large  powers,  must, 
for  good  results,  be  strongly  laced  by  one  of  the  methods 
already  given,  or  an  equally  good  one.  Among  the 


d 


(MM 


FIG.  41. 


various  metallic  belt-hooks  we  may  give  the  following 
as  probably  the  best  in  use :  Fig.  41  represents  a  kind 
of  belt-hook  which  is  quite  extensively  used  for  light 
belts.  Figure  a  is  the  hook  itself.  To  fasten,  proceed 
as  follows :  Cut  slits  in  the  belt-ends  parallel  in  length 
to  the  length  of  the  belt ;  place  the  ends  as  shown  in 
Fig.  b ;  force  through  the  slits  the  belt-hooks  as  in  the 
figure,  turn  them,  and  flatten  out  the  belt  as  in  figure  c. 


LACING  AND    OTHER  MODES   OF  FASTENING.      73 

Figure  d  represents  the  pulley-side   of   the  belt  and 
figure  c  the  outside. 

In  Fig.  42  the  hook  (figure  a)  has  a  double  hold  on 
the  belt  through  the  two  rows  of  holes,  and  is  there- 
fore a  stronger  fastener  than  the  preceding  hook. 
Figure  b  represents  the  outside  of  a  belt  fastened  with 


n  n 


n  n  n 


FIG.  42. 

these  hooks,  figure  c  the  pulley-side,  and  figure  d  a 
section  through  the  two  ends  of  the  belt  showing  one 
hook. 

An  ingenious  buckle  for  fastening  together  the  belt- 
ends  is  given  in  Mr.  Cooper's  "  Use  of  Belting,"  and 
credited  to  a  Canadian  inventor.  The  fastener  consists 


74 


BELTS  AND  PULLEYS. 


of  two  separate  parts,  one  containing  a  series  of  parallel 
metallic  tongues  (represented  by  the  dotted  lines  in 
figure  43  a)  which  are  inserted  through  holes  in  the 
belt-ends,  and  the  other  a  rectangular  cover  which  is 
slipped  over  the  projecting  ends  of  the  tongues  after 
they  have  been  forced  through  the  belt.  Figure  43  a 
represents  the  outside  and  figure  b  the  pulley-side  of 
the  belt.  Figure  c  is  a  sectional  drawing  showing  a 
pair  of  tongues  and  the  cover. 


a 


FIG.  43. 

All  belt-hooks  and  metallic  fasteners  used  for  belts 
to  be  run  over  pulleys  should  be  of  brass,  copper,  or 
other  soft  metal,  in  order  to  prevent  scratching  the 
surface  of  the  pulley,  and  the  consequent  additional 
wear  of  the  whole  belt. 

A  very  simple,  if  not  very  firm  and  secure,  method 
of  fastening,  without  the  use  of  lacing  thongs  or  hooks 
of  any  kind,  is  shown  in  Fig.  44.  One  end  of  the  belt 


STRENGTH  OF  LEATHER  BELTS. 


75 


is  cut  into  cleat-shaped  pieces,  shown  in  figure  b  at 
y,  y->  y>  and  the  other  punched  with  oblong  slots,  figure 
a,  x,  x,  x.  The  cleats  are  forced  through  the  slots,  the 
belt-ends  hammered  out  flat,  and  the  joining  is  complete. 
Figure  c  shows  a  section  through  the  ends  of  the  belt, 


FIG.  44. 

with  the  cleat  and  slot  fastening.  Such  a  fastening  as 
this  is  at  best  weak  and  uncertain,  and  should  not  be 
used  at  all  in  practice,  except  for  some  exceptionally 
light  work,  where  lacing-thongs  or  belt-hooks  are  not 
easily  to  be  obtained. 


§  10.     Strength  of  Leather  Belts — Resistance  to  Slipping. 

The  discussion  of   the  strength  and   resistance   to 
slipping  of  leather  belts  is  attended  with  well-nigh  in- 


76  BELTS  AND  PULLEYS. 

surmountable  difficulties,  from  the  fact  that  the  sub- 
stance with  which  we  have  to  deal  is  almost  wanting 
in  homogeneity.  We  are  able  by  means  of  standard 
rules  and  formulas  to  calculate  closely  the  strength  of 
a  cast-iron  column  or  wrought  girder,  because  within 
reasonable  limits  cast-iron  and  wrought-iron  are  homo- 
geneous ;  in  other  words,  if  we  know  the  breaking 
strength  and  safe-working  strength  of  a  certain  kind  of 
iron,  we  know  these  strengths  of  other  iron  of  the  same 
kind:  they  are  approximately  the  same.  Other  metals 
also  are  even  in  texture  and  homogeneous  in  nature ; 
many  kinds  of  wood  possess  this  valuable  homo- 
geneity to  a  remarkable  extent.  But  this  is  by  no 
means  true  of  leather.  Few  substances,  if  any,  with 
which  mechanical  men  have  to  deal  show  such  widely 
varying  results  under  apparently  similar  circumstances 
as  the  leather  which  furnishes  for  us  the  countless 
number  of  transmission-belts  seen  in  nearly  every  shop 
and  factory  in  the  land.  In  a  series  of  tests  made  by 
a  prominent  firm  of  leather-belt  manufacturers  in  New 
York  City,  strips  of  leather  two  inches  wide  were  cut 
from  one  of  the  ordinary  sides  used  for  belting,  and 
carefully  tested  in  the  same  testing-machine  and  under 
precisely  similar  circumstances.  These  strips  were 
broken  at  strains  varying  all  the  way  from  1400  pounds 
to  3475  pounds  ;  which  result  elicits  the  strange  fact, 
that  one  strip  of  leather  may  be  nearly  two  and  a  half 
times  as  strong  as  another  strip  equal  in  width  and 
thickness,  and  taken  from  the  same  side  of  leather. 
The  strips  in  question  when  in  their  original  positions 
in  the  skin  were  but  15  inches  apart  at  their  nearest 
points.  Nor  is  this  all :  in  two  strips  which,  in  the 


STRENGTH  OF  LEATHER  BELTS.  77 

side  of  leather,  joined  each  other,  lay  immediately  side 
by  side,  the  difference  in  breaking  strength  was  675 
pounds,  or  33/2"  pounds  per  inch  of  width  ;  a  variation 
of  32  per  cent  of  the  greater  strength  and  of  nearly  47 
per  cent  of  the  smaller. 

A  gentleman  for  many  years  engaged  in  the  manu- 
facture of  leather  belting  has  informed  the  author  that 
he  once  cut  off  twelve  inches  of  solid  part  (i.e.,  without 
rivets  or  splicing)  from  a  roll  of  two-inch  belting ;  cut 
the  piece  longitudinally  into  two  parts ;  tested  them 
in  a  correct  machine ;  and  found  that  one  part  with- 
stood 400  pounds  greater  tensional  strain  than  the 
other.  The  gentleman  also  said  that  he  had  tested 
with  a  good  dynamometer  two  eight-inch  belts,  made 
from  similar  leather  in  his  own  factory,  running  over 
pulleys  equal  in  size,  doing  the  same  kind  of  work,  and 
carefully  stretched  over  their  pulleys  with  as  nearly  as 
possible  the  same  tensions,  and  found  that  one  would 
transmit  nearly  a  horse-power  more  work  without  slip- 
ping than  the  other.  Many  other  similar  examples 
from  practice  might  be  cited  to  show  with  how  much 
pf  uncertainty  and  variation  from  averages  the  investi- 
gator of  belt-transmissions  is  compelled  to  deal.  Let 
the  examples  already  given,  however,  suffice  for  this 
purpose ;  and  let  us,  keeping  always  well  on  the  safe 
side,  endeavor  to  calculate,  as  simply  as  the  compli- 
cated nature  of  the  subject  will  allow,  the  proper 
strengths  and  sizes  for  the  various  transmission-belts  in 
use  in  practice. 

The  strain  brought  to  bear  upon  an  ordinary  endless 
belt  running  continuously  over  its  pulleys,  leaving  out 
of  the  question  considerations  due  to  centrifugal  force, 


78  BELTS  AND  PULLEYS. 

etc.,  etc.,  is  one  of  simple  tension  ;  and  were  it  not  for 
other  complicating  elements  which  enter  into  the  cal- 
culations, the  proper  strength  for  a  belt  to  withstand 
a  certain  strain  could  be  quite  easily  calculated.  For 
example,  if  we  represent  by  P  the  actual  strain  on  the 
belt  in  pounds,  by  A  the  cross-section  of  the  belt  in 
square  inches,  and  by  f  the  safe  working  tensional 
stress  in  pounds  per  square  inch  for  the  material  of  the 
belt,  we  can  write  the  formula 

P=Af, 


and,  by  transposing,      A  =  -^. 


From  this  simple  formula,  were  the  tensional  strain 
all  which  we  must  take  into  account,  we  could  easily 
calculate  our  belt  widths  and  thicknesses.  But,  un- 
fortunately for  the  simpli- 
city of  our  calculations, 
other  considerations  must 
be  looked  into  before  we 
can  correctly  obtain  the 
necessary  rules  and  formu- 
las. In  the  first  place, 
probably  nine  belts  out  of 
ten  in  ordinary  use  will  slip 
around  on  their  pulleys 
FIG  before  they  will  break ;  that 

is,  the  resistance  of  the 
belt  to  slipping  is  not  equal  to  its  strength.  It  there- 
fore becomes  necessary  to  embody  in  our  calculations 


STRENGTH  OF  LEATHER   BELTS.  79 

for  strength  considerations  which  will  prevent  slipping 
of  the  belt  upon  its  pulleys. 

Let  ACB  (Fig.  45)  represent  a  band  or  cord  drawn 
over  an  angle  of  a  solid,  as  shown  in  the  figure.  Let 
forces,  represented  by  T  and  /,  act  at  the  ends  of 
the  cord  in  the  directions  shown,  and  let  a  represent 
the  angle  DCB.  In  drawing  the  cord  over  the  angle 
or  corner  the  friction  between  the  block  and  cord 
must  be  overcome.  By  the  principles  of  the  parallelo- 
gram of  forces,  the  resultant  normal  pressure  R  of  the 
forces  T  and  /  is  given  by  the  expression 


R  =  Vr  +  ?—2~Ttcos<x;      ....     (33) 

and  if  we  represent  by  F  the  friction  and  by  cp  the 
coefficient  of  friction,  we  shall  have 


F  =  <pR  =  <p  V'T*  -\-f~-2Tt  cos  a. 

In  order  to  move  the  cord  over  the  angle  in  the  direc- 
tion of  the  force  T,  this  force  must  be  able  to  over- 
come the  force  t  acting  in  an  opposite  direction,  and 
also  the  friction  ,  that  is,  we  must  have 


From  this,  by  squaring, 

r2  =  f  +  2tF  '+  F\ 

Substituting  this  value  of  7"2  in  the  above  equation  for 
the  friction,  and  neglecting  the  quantity  F2,  gives  us  the 
equation 

F  =  <p  Vf  +  2tF-\-  f  —  2f  cos  a  —  2Ft  cos  a 


SO  BELTS  AND  PULLEYS. 

and  by  factoring  we  obtain 


F=  y  4/2(1  -  cos  a)  (f  +  tF). 
From  trigonometry  we  find 


—  cos  a)  =  sin  %a, 
which,  multiplied  by   4/4,  becomes 

4/f(i  —  cos  a)  =  sin  \ot  4/4, 


or  4/2(1  —  cos  a)  =  2  sin  \ot. 

Consequently 

F  —  2cp  sin  -  4/?  +  /F. 

From  the  binomial  formula,  neglecting  the  small 
terms  after  the  second,  we  may  extract  the  square 
root  of  the  quantity  under  the  radical  sign,  and  write 


~. 


Hence  F=  2q>  sin -\* -r-      /, 

2  / 

F=  2<pt  sin  — \-  q>F  sin  -, 

2  2J 

a  a 

r  —  CDF  sm  —  =  2q)t  sin  — , 

2  2 


STRENGTH  OF  LEATHER  BELTS.  8 1 


.     a 

sm  - 

and  finally  F=  -  — (34) 

i  —  (p  sin  — 

The  force,  then,  which   is  required  to  draw  the  cord 
over  the  angle  in  the  direction  of  T  is 

a 

sm  — 


.     a' 
l  -  cpsm- 


or  T=t 


When  the  angle  a  is  very  small  we  may  say  correctly 
enough 

a 
I  —  (p  sin  —  =  i, 

and  formula  (35)  becomes 

+  2<f>Sin^ (36) 

Suppose  now  instead  of  one  angle  over  which  to 
draw  the  cord  we  have  several,  as  shown  in  Fig.  46,  the 
angles  being  equal  each  to  each.  Let  /  be  the  tension 
at  one  end  of  the  cord,  tl  that  at  the  first  angle,  /a  that 
at  the  second  angle,  etc.,  to  the  tension  T  =  tn  at  the 
other  end.  From  what  precedes,  we  shall  have  for  the 
force  necessary  to  draw  the  cord  over  the  first  angle 


82 


BELTS  AND  PULLEYS. 


ti  =  t  (l  -f-  2cp  sill  -). 


For  the   force    necessary  to  draw  the  cord   over  the 
second  angle  we  shall  have 


Hence 


or 


sin         i  +  2cp  sin 


/  «\a 

=  /  ^i  +  2y  sm  -J  . 


FIG.  46. 


In  a  similar  manner 


=^        I 


•    a\ 
sin  -j, 


<x\*r  a\         f  .    fx\3 

sin  -      i  +  2<p  sm  -J  =  /  ( I  +  2q>  sm  -  J 
^/  \  ^&/          \  */ 


STRENGTH  OF  LEATHER  BELTS. 


And  finally 


a 


tn=  T  —  t  ( i  -f  2cp  sin  - 


•    (37) 


By  means  of  this  formula  we  are  able  to  calculate 
the  forces  which  tend  to  cause 
an  endless  belt  to  slip  upon  its 
pulley,  the  tensions  in  the  belt 
necessary  to  prevent  slipping, 
and  consequently  the  strength 
and  width  of  the  belt  itself. 

Let  K,  Fig.  47,  be  a  pulley,  A 
over  which,  embracing  a  centre 
angle  BCA  =  EDB  =  a,  a  belt 
tABT  passes  as  shown  in  the 
figure.  We  can  assume  the  arc 
AB  to  be  composed  of  an  infinite  number  (n)  of  in- 

finitely small  sides  ;  each  will  then  be  expressed  by  —  . 

it 

From  formula  (37)  we  have  for  the  force  T  the  ex- 
pression 


FIG.  47. 


/  a  \n 

=.  t    i  4-  2<p  sin  —    , 

V  2H)    ' 


and  since  for  the  infinitely  small  arcs  their  sines  are 

equal  to  the  arcs  themselves,  we  may  write,  sin  —  =  — , 

2n       2n 

and  therefore 


This  expression  we  may  develop  by  means  of  the  bi- 
nomial theorem  into 


84  BELTS  AXD   PULLEYS. 


n 


v 
I  X2X  3^3  °V; 

and  since  we  have  assumed  n  to  be  infinitely  great,  we 
may  write  n  —  i  =  n  —  2  =  n  —  3  =  n.  Our  last 
equation  therefore  becomes 


This  is  in  the  form  of  the  series 


in  which  e  represents  the  base  of  the  Naperian  or  hyper- 
bolic system  of  logarithms  (e  —  2.71828),  and  the  above 
equation  reduces  to 


(38) 
From  this  we  have 

hyp.  log  T  --  hyp.  log  /  =  (pa, 

T 
and  *  hyp.  log  --  =  cpa  .....     (39) 

*  This  may  be  very  neatly  demonstrated  by  means  of  the  integral  cal- 
culus as  follows:  Let  a.  represent  the  entire  arc  embraced  by  the  belt, 

and  d.  a.  one  of  the  small  portions  which  we  represented  above  by—. 
The  tension  at  the  point  A  is  /,  that  at  the  next  portion  of  arc  i-\-d.t\ 


STRENGTH   OF  LEATHER  BELTS.  85 

Common  logarithms  are  better  known  and  more 
easily  handled  than  the  hyperbolic.  To  reduce  for- 
mula (39)  to  common  logarithms  it  is  necessary  only  to 
multiply  by  0.434.  Thus 

T 

!°gy  —  0.4349*, (40) 

where  a  is  expressed  in  circular  measure,  i.e.,  parts  of  n. 
If  a  is  taken  in  degrees,  substitute  a  =  -~— ,  and  we 
obtain 

T 

log  --  =  0.0075789^ (41) 

If  a  is  taken  in  fraction  of  the  circumference,  sub- 
stitute a  =  2ita.  We  obtain  thus 

T  _ 
t 


the  increase  is  d.t,  and  this  is  due  to  the  friction  in  the  unit  of  arc. 
This  friction  is  d.F  =  %cpt  sin   -— - ;   or,   since  d.a  is  very  small, 

d.F  —  2(pt—-  —  cptct.a. 

Hence  we  have     d.t  —  cptd.a       or      — -  —  cpd.a. 
Integrating  between  the  limits  7" and  /,  a  and  o,  gives  us 

fTd.t  /*«  T 

I     -—  =  cp  I    d.a,     or      hyp.  log  -  —  cpa. 

tJt  ?  t/Q  » 


86  BELTS  AND   PULLEYS. 

The  best  and  most  recent  experiments  made  use  of 
for  determining  the  coefficient  of  friction  permit  us  to 
use,  for  ordinary  belt-leather  over  cast-iron  pulleys,  the 
value 

*  V  =  o-4 (43) 

This  value  substituted  in  formula  (40)  gives 


T 

log  —  =  0.434  X  04**, 


or,  when  a  is  in  circular  measure, 


T 

log-    =  o.i  736** (44) 


Substituting  q>  —  04  in  formula  (41)  gives 

T 
log  -    =  0.007578  X  040:, 


or,  when  a  is  in  degrees, 

T 

log  -  =  0.00303^ (45) 


fSee  Appendix  I. 


STRENGTH  OF  LEATHER  BELTS.  8/ 

Similarly,  by  substituting  in  formula  (42), 

T 

log  7-  =  2-729  X  0.4^, 

or,  where  a  is  a  fraction  of  the  circumference, 


T 

log  -  =  1.0916** (46) 

t 


The  following  table,  calculated   from  formulas  (44), 

T 

(45),  and  (46),  gives  values  of  --  for  different  values  of 

the  arc  a  from  30°  to  300°  corresponding  to  from  0.524 
to  5.236  in  circular  measure,  and  from  T^-  =  0.083  to 
-|  =  0.833  in  fractions  of  the  circumference. 

To  illustrate  the  application  of  the  table,  suppose 
we  have  a  pair  of  cast-iron  pulleys  over  which  we  pro- 
pose to  run  a  leather  belt.  Suppose  the  arc  embraced 
by  the  belt,  upon  the  pulley  over  which  it  is  most 
likely  to  slip  (the  pulley  having  the  smaller  amount  of 
contact  with  the  belt,  or  the  smaller  pulley),  is  75°  = 
1.309  in  circular  measure  —  ^  —  0.208  in  fraction  of 
the  circumference. 

We  look  along  the  column  of  degrees  until  we  find 
the  value  75°,  along  the  column  of  circular  measures 
until  we  find  1.309,  or  along  the  column  of  fractions 
of  the  circumference  until  we  find  -/^  =  0.208,  and,  op- 
posite to  these  values  we  find  the  required  value  for 

T 

the  ratio  of  the  tensions,   —  =  1.689. 


BELTS  AND  PULLEYS. 


TABLE  OF  TENSIONS  FOR  LEATHER  BELTS  OVER  CAST-IRON 
PULLEYS. 


a  = 

T 

In  degrees. 

In  circular 
measure. 

In  fractions  of  the 
circumference. 

t 

30 

0.524 

tV  =  °'°83 

1.233 

45 

0.785 

i  =  0.125 

1.369 

60 

1.047 

\  =  0.167 

1.521 

75 

1.309 

•ft  —  0.208 

1.689 

90 

I-57I 

i  =  0.250 

1.874 

105 

1.833 

A  =  0.292 

2.082 

1  20 

2.094 

*  =  0.333 

2.312 

135 

2.356 

1  =  0.375 

2.565 

150 

2.618 

A  =  0.417 

2.849 

165 

2.880 

tt  =  0.458 

3.163 

180 

3.142 

|  =  o  500 

3.5I4 

195 

3.403 

M-  0.541 

3.901 

210 

3.665 

A  =  0-583 

4-333 

240 

4.189 

|  =  0.667 

5.340 

27O 

4.712 

f  =  0.750 

6.589 

300 

5-236 

f  =  0.833 

8.117 

The  greatest  strain  brought  to  bear  upon  an  endless 
belt,  or  the  strain  tending  in  the  greatest  degree  to 
cause  breakage,  is  the  tension  in  the  driving  part  of 
the  belt,  that  is  T.  This  tension  acts  in  one  direction 
and  the  lesser  tension  /  in  a  contrary  direction.  Con- 
sequently it  is  the  excess  of  the  greater  over  the  lesser 
tension  which  overcomes  the  resistance  of  the  pulley 
and  causes  rotation.  If  we  represent  the  force  of  re- 
sistance in  pounds  at  the  circumference  of  the  pulley 
(which  is  the  force  transmitted  by  the  pulley)  by  P9  we 
shall  have  the  expression 


P=  T  -  t. 


(47) 


STRENGTH   OF  LEATHER  BELTS.  89 

Hence  T  =  P  +  t, 

which  may  be  put  in  the  form 


By  substituting  for  P  within  the  parenthesis  its  value 
from  formula  (47),  we  obtain 


But 


T-  t"  T 

T 


Hence  T  =  Pi  l  +  ~— -  \; 


.T 

----!  +  ! 


T 
T=P{      —-}, (48) 


9o 


BELTS  AND  PULLEYS. 


by  means  of  which  and  the  preceding  table  the  ten- 
sion T  for  different  values  of  a  may  be  determined. 
The  following  table,  calculated  from   formula  (48), 
T_ 

gives  values  of 


T 


-  for  different  values  of  the  arc  a. 


TABLE  OF  GREATEST  TENSION  FOR  LEATHER  BELTS  OVER  CAST-IRON 

PULLEYS. 


In  degrees. 

In  circular 
measure. 

In  fractions  of  the 
circumference. 

J  —  /'X 

30 

0.524 

rV  =  0.083 

5.29 

45 

0.785 

i  =  0.125 

3.71 

60 

1.047 

\  =  0.167 

2.92 

75 

1.309 

-^  =  0.208 

2.45 

90 

I-57I 

I  =  0.250 

2.14 

105 

1.833 

A  =  0.292 

1.93 

120 

2.094 

i  =  0-333 

1.77 

135 

2.356 

1  =  0.375 

1.64 

150 

2.618 

A  =  0.417 

1.54 

I65 

2.880 

4i  =  0.458 

1.47 

1  80 

3.142 

i  -  0.500 

1.40 

195 

3.403 

H  =  0.541 

1.35 

2IO 

3.665 

TV  =  0.583 

1.30 

24O 

4.189 

f  =  o.667 

1.23 

27O 

4.712 

I  =  0.750 

1.18 

300 

5-236 

1  =  0.833 

1.14 

To  illustrate  the  use  of  the  table :  Suppose  the  force 
transmitted  by  a  pulley  is  P  —  500  pounds  and  the  angle 
embraced  by  the  belt  a  =  105°.  In  the  table  opposite 
to  the  value  a  =  105°  we  find  the  value  1.93.  Hence 
T  —  P  X  1.93  =  500  X  1.93  or  T  —  965  pounds. 


STRENGTH  OF  LEATHER  BELTS.  gi 

We  have  now  developed  rules  by  which  the  actual 
strain  upon  the  belt  may  be  determined  :  we  have  still 
to  determine  the  strength  of  the  belt,  or,  in  other 
words,  the  amount  of  material  necessary  in  the  belt  to 
safely  sustain  the  given  strain.  We  have  said  that  the 
strain  T  upon  an  endless  belt  is  a  tensional  strain.  If, 
therefore,  we  represent  by  b  the  breadth  of  the  belt  in 
inches,  by  8  its  thickness,  also  in  inches,  and  by  /the 
greatest  safe-working  stress  in  pounds  per  square  inch, 
we  shall  have,  for  the  relation  between  the  strain  and 
the  strength,  the  expression 


(49) 


T 

and  consequently        bS  =    TT  ........     (50) 

Because  of  the  great  variations  in  the  strength  of 
leather  the  quantity  f  can  be  only  approximately  de- 
termined. Experiments  and  tests  upon  the  strength 
of  leather,  be  they  ever  so  numerous  and  carefully 
made,  serve  only  to  impress  more  strongly  upon  the 
mind  of  the  experimenter  this  unfortunate  lack  of  ho- 
mogeneity in  the  substance  with  which  he  is  dealing. 
In  this  predicament  he  who  would  investigate  the  sub- 
ject of  leather  belts  must  be  satisfied  with  an  average 
value  taken  from  a  great  many  widely  differing  values 
for  his  coefficient  of  strength  ;  and  until  our  manufac- 
turers are  able  to  produce  leather  which  shall  be  to  a 
reasonable  extent  -uniform,  the  subject  of  strength  of 
belting  must  remain  as  it  is  now  —  the  most  uncertain 
and  indefinite  one  with  which  mechanical  men  have 
to  deal. 


92  BELTS  AND  PULLEYS. 

The  weakest  part  of  an  endless  belt  is  obviously  at 
the  joint:  the  value  of  the  safe-working  stress /must 
therefore  be  taken  for  this  part.  The  author  has  dur- 
ing the  last  three  years  tried  a  great  many  experiments 
with  the  view  of  obtaining  the  average  strength  of 
laced  and  riveted  joints.  These  average  breaking 
strengths  he  has  found  to  be  about  as  follows : 


For  ordinary  single  leather-lacing, 

950  pounds  per  square  inch  ; 
For  ordinary  single  rawhide-lacing, 

1000  pounds  per  square  inch  ; 
For  good  double  leather-lacing, 

1 200  pounds  per  square  inch ; 
For  good  double  rawhide-lacing, 

1400  pounds  per  square  inch  ; 
For  ordinary  riveted  joints, 

1750  pounds  per  square  inch. 

We  may  therefore  take  for  our  safe-working  stress 
in  pounds  per  square  inch  the  following  values : 

Single  leather-lacing,  /  —  325  ; 
Single  rawhide-lacing,  f=  3 So; 
Double  leather-lacing,  f  =  375  ; 
Double  rawhide-lacing,  f  =  400  ; 
Riveted  joints,  f=  575. 

By  substituting  these  values  successively  in  formula 
(50),  we  obtain  the  following  formulas : 


STRENGTH   OF  LEATHER  BELTS.  93 

f 

For  single  leather-lacing,       66  =    — ;.     .     .     .     (51) 


f 

For  single  rawhide-lacing,      66  = ; .     .     .     .     (52) 


T 
For  double  leather-lacing,     66  =  -— ;  .     .     .     .     (53) 


T 
For  double  rawhide-lacing,    66  = ;  .     .     .     .     (54) 


T 

For  a  riveted  joints,  66  =  --— (55) 


Example. — Required  the  width  of  a  leather  belt  J 
inch  thick,  which  will  safely  transmit  a  force  of  P  =  600 
pounds  when  laced  according  to  each  of  the  above-men- 
tioned methods,  the  pulleys  over  which  the  belt  is  to 
run  being  of  the  same  diameter — that  is,  the  angle  em- 
braced by  the  belt  being  a  —  180°. 

From  the  table  on  page  90  we  have,  T  =  P  X  1.40 
=  600  X  1.40  =  840  pounds.  From  formula  (5.1), 
therefore,  we  have 

i         840  4  X  840 

*X4==3*?  -355-' 

or,  for  single  leather-lacing, 

b  =  10.34"  =  IOU". 


94  BELTS  AND  PULLEYS. 

From  formula  (52), 

I        840  4  X  840 


or,  for  single  rawhide-lacing, 

b  =  9.6"  =  9H". 
From  formula  (53), 

i        840         t  _  4  X  840 
^=375'  ~37T~' 

or,  for  double  leather-lacing, 

*  =  8.96"  =  8ff'  . 

From  formula  (54), 

i     _  840  4  X  840 

U     X     '  —  ,  6?    -     - 

4        400  400 

or,  for  double  rawhide-lacing,     * 

b  =  8.40"  =  8i|r/. 
From  formula  (55), 

/,  ^  l     -  84°          T,  _  4  X  840 
^4       57?  ^?T 

or,  for  a  riveted  joint, 

*  =  5.84"  =  sir- 


STRENGTH  OF  LEATHER  BELTS.  95 

The  following  tables  of  formulas  have  been  calculated 
from  the  table  on  page  90  and  formulas  (51),  (52),  (53), 
(54),  and  (55),  respectively.  The  above  example  may 
be  calculated  from  these  tables  as  follows  :  We  have 
for  our  data,  P  =  600  pounds,  a  =  180°,  and  d  =  J". 
From  formula  (66),  for  single  leather-lacing, 

bS  =  0.0043 1  X  600 ; 
b  =  0.00431  X  600  X  4  =  10.34". 

From  formula  (82),  for  single  rawhide-lacing, 

b$  —  0.004  X  600 ; 
b  —  0.004  X  600  X  4  =  9.60". 

From  formula  (98),  for  double  leather-lacing, 

bd  —  0.00373  X  600 ; 
b  —  0.00373  X  600  X  4  —  8.952". 

From  formula  (114),  for  double  rawhide-lacing, 

bd  —  0.0035  X  600 ; 
b  —  0.0035  X  600  X  4  =  8.4o"« 

From  formula  (130),  for  a  riveted  joint, 

bd  =  0.00243  X  600 ; 
b  —  0.00243  X  600  X  4  =  5.832". 


96 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  CAST-IRON  PULLEYS. 
Single  Leather  Lacing. 


a.  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A-  =  °-°83 

bd  =  O.OI628/* 

56 

45 

0.785 

i  =  0.125 

bd  =  O.OU42/* 

57 

60 

1.047 

\  =  0.167 

bd  =  o.ooSgS/3 

58 

75 

1.309 

&  =  0.208 

bd  =  0.00754/* 

59 

90 

I-57I 

J  =  0.250 

bd  =  o.oo658/> 

60 

105 

1.833 

&  =  0.292 

b8  =  0.00594^ 

6l 

I2O 

2.094 

i  =  0.333 

3<5  =  0.00545^ 

62 

135 

2.356 

t  =  0.375 

bd  =  0.00505/5 

63 

150 

2.618 

^  =  0.417 

bd  =  0.00474^ 

64 

I65 

2.880 

tt  =  0.458 

3<5  =  0.00452/* 

65 

180 

3.142 

-J-  =  0.500 

3<5  =  0.0043I/3 

66 

J95 

3.403 

tt  =  °-54i 

3<5  =•  0.00415/5 

67 

210 

3.665 

S  =  o.583 

^6  =  0.00400^ 

68 

240 

4.189 

t  =  o.667 

bd  =  0.0037S/5 

69 

270 

4.712 

f  =  0.750 

^5  =  0.00363/3 

70 

300 

5.236 

1-0.833 

^5   rr  0.0035I.P 

71 

TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  CAST-IRON  PULLEYS^ 
Sing  le  Ra  wh  ide  La  dug. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

^  =  0.083 

35  =  O.OI5II/* 

72 

45 

0.785 

i  =  0.125 

bd  =  o.  01060  P 

73 

60 

1.047 

J  =  0.167 

bd  =  0.008347* 

74 

75 

1.309 

•£t  =  0.208 

bd  =  o.oojooP 

75 

90 

I.57I 

}  =  0.250 

bd  =  o.oo6n/> 

76 

105 

1.833 

¥7T  =  0.292 

bd  —  0.005  5  1/3 

77 

120 

2.094 

*  =  0-333 

35  =  o.  00506  />  . 

78 

135 

2.356 

f  =  0.375 

bd  =  0.00469^ 

79 

150 

2.618 

A—  0.417 

bd  =  O.OO44O/* 

80 

165 

2.880 

=0.458 

bd  =  O.OO42O/* 

Si 

180 

3.142 

i  =  0.500 

bd  =  0.00400/3 

82 

*95 

3.403 

J}  =  0.541 

bd  =  0.00386/5 

83 

210 

3.665 

T¥  =  0.583 

bd  =  O.OO37I/3 

84 

240 

4.189 

£  =  0.667 

3(5  =  0.0035I/3 

85 

270 

4.712 

f  -  0.750 

36X  =  0.00337/* 

86 

300 

5.236 

1  =  0.833 

3<5  =  O.OO326/* 

87 

f. 


STRENGTH  OF  LEATHER  BELTS. 


N7- 

97 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  CAST-IRON  PULLEYS. 
Double  Leather-Lacing* 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

Y1^  =  0.083 

35  =  O.OI^llP 

88 

45 

0.785 

i  =  0.125 

bd  —  0.00989^ 

89 

60 

1.047 

t  =  0.167 

35  =  0.007  79  /* 

90 

75 

1.309 

•&  =  0.208 

35  =  O.OO653/* 

91 

QO 

I-57I 

i  =  0.250 

35  =  0.0057I/3 

92 

105 

1.833 

•fa  =  0.292 

35  =  0.005I4/* 

93 

120 

2.094 

i  =  0.333 

35  =  o.  00472^ 

94 

135 

2.356 

f  =  0.375 

35  =  0.00437^ 

95 

150 

2.618 

-fz  =  0.417 

^  =  0.0041  \P 

96 

165 

2.880 

tt=  0.458 

bd  =  0.00392/* 

97 

180 

3.142 

|  =  0.500 

b8  =  0.00373/5 

98 

195 

3.403 

if  =  0.541 

35  =  o.  00360  P 

99 

210 

3.665 

X=  ,9.583 

bd  =  0.00347/* 

IOO 

240 

4.189 

%  —  0.667 

bd  =  0.00328/* 

101 

270 

4.712 

f  =  0.750 

^5  =  O.OO3I5/* 

102 

300 

5-236 

f  =  0.833 

^5  =i  0.00304/* 

103 

TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  CAST-IRON  PULLEYS. 
Double  Rawhide- Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

bd  =  O.OI323/3 

104 

.    45 

0.785 

i  -  0.125 

bd  =  o.  00928^ 

105 

60 

1.047 

i  =  0.167 

bd  =  o.  00730^ 

1  06 

75 

1.309 

-ff  =  0.208 

bd  =  o.  00613  P 

107 

90 

I-57I 

J  =  0.250 

bd  —  0.00535^ 

108 

105 

1.833 

•ff  =  0.292 

M  =  o.  00483  P 

109 

120 

2.094 

i  =  0.333 

bd  =  O.OO443/* 

no 

.135 

2.356 

1  =  0.375 

^5  —  O.OO4IO/* 

in 

150 

2.618 

fV  =  0.417 

bd  =  0.00385/3 

112 

I65 

2.880 

fl-  =  0.458 

35  =  0.00368/5 

H3 

1  80 

3.142 

|  —  0.500 

35  =  O.OO35O/5 

114 

195 

3.403 

Jf  =  0.541 

35  —  0.00338/5 

U5 

2IO 

3-665 

A  =  0.583 

35  =  o.  003257* 

116 

240 

4.189 

f  =  0.667 

35  =  0.00308/* 

117 

270 

4.712 

I  =  0.750 

35  =  0.00295/3 

118 

300 

5.236 

f  -0.833 

35  =  0.00285^ 

119 

98 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  CAST-IRON  PULLEYS. 
Riveted  Joint. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

•^  ==  0.083 

bd  =  0.009207* 

1  20 

45 

0.785 

t  =  0.125 

M  =  0.00645/* 

121 

60 

1.047 

t  =  0.167 

^5  =  o.oosoS/* 

122 

75 

1.309 

•ff  =  0.208 

^5  =  0.00426P 

J23 

90 

I-57I 

i  =  0.250 

bd  =  0.00372/* 

124 

105 

1.833 

^=0.292 

bd  =  0.00336/* 

125 

1  20 

2.094 

t  =  0-333 

^d  =  0.003087* 

126 

135 

2.356 

1  =  0-375 

M  =  o.  002857* 

127 

150 

2.618 

A  =  0.417 

^^  =  0.00268/3 

128 

165 

2.880 

tt  =  °-458 

^5  =  0.002567* 

129 

180 

3.142 

|  —  0.500 

b8  =  0.00243  P 

130 

195 

3.403 

H  =  0.541 

bd  —  0.002357* 

131 

210 

3-665 

&  =  0.583 

b8  =  0.002267* 

132 

240 

4.189 

f  =  o.c67 

b8  =  0.002147* 

133 

270 

4.712 

t  =  0.750 

^5  =  0.002057* 

134 

300 

5-236 

1  =  0.833 

b8  =  0.001987* 

135 

STRENGTH  OF  LEATHER  BELTS.  99 

Often,  when  we  know  the  horse-power  to  be  trans- 
mitted, it  is  convenient  to  calculate  belt-widths  from 
this,  without  rinding  the  circumferential  force.  From 
formula  (20)  we  have,  when  vm  represents  the  velocity 
in  feet  per  minute,  and  H  the  horse-power, 


(I36) 

^    *    } 


and  from  formula  (22),  when  v  represents  the  velocity 
in  feet  per  second, 


By  substituting  this  last  value  of  P  in  formulas  (56) 
to  (135),  and  reducing,  we  may  obtain  the  following 
tables  of  formulas  for  calculating  belt-widths  from  the 
horse-power  transmitted  and  the  velocity  in  feet  per 
second  :* 

'  *  By  substituting  the  value  of  P  given  in  formula  (136)  in  formulas 
(56)  to  (135),  we  may  obtain  formulas  for  belt-widths  in  terms  of  the 
horse-power  and  velocity  in  feet/<?r  minute.  For  example,  formula 

(68)  gives  b<$  =  0.004  —      —  =  132 — .       Such    formulas   are,   how- 

m  m 

ever,    seldom   needed  in  practice,  the  velocity  being  almost  always 
taken  in  feet  per  second. 


100 


BELTS  AND  PULLEYS. 


TABLE  OF   FORMULAS   FOR   LEATHER-BELTS   OVER  CAST-IRON 
PULLEYS. 

Single  Leather-Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

bd  •=  8.954  — 

138 

45 

0.785 

i  =  0.125 

V 

139 

60 

1.047 

1  =  0.167 

ff 

bd  =  4.939- 

140 

75 

1.309 

•fa  =  0.208 

bd  =  4.147^ 

V 

141 

90 

I.57I 

i  =  0.250 

bd  =  3.619^ 

V 

142 

105 

1-833 

•h  =  0.292 

H 

bd'  =  3.267  — 

143 

1  20 

2.094 

i  =  0.333 

TT 

bd  =  2.998  — 

144 

135 

2.356 

1  =  0.375 

bd  =  2.778^ 

145 

150 

2.618 

A  =  0.417 

^5  =  2.607  — 
v 

146 

165 

2.880 

M-  0.458 

M  =  2.486^ 
» 

147 

1  80 

3.142 

i  =  0.500 

*d  =  *v37if 

148 

195 

3.403 

tt  =  0.541 

^  =  2.283^ 
» 

149 

2io 

3.665 

A  -0.583 

,.               ^ 

/7O   =  2.2OO  — 
0 

% 

240 

4.189 

1  =  0.667 

£<$  =   2.O7Q— 

Z/ 

151 

27^) 

4.712 

£  -  0.750 

IX                          H 

bo  =  1.997  — 

152 

300 

5-236 

1  =  0.833 

H 

00  =  1.931  — 

v 

153 

STRENGTH  OF  LEATHER  BELTS. 


IOI 


TABLE  OF  FORMULAS  FOR  LEATHER-BELTS  OVER  CAST-] 
PULLEYS. 

Single  Raw  hide- Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0 

524 

A 

=  0.083 

bd 

=  8 

H 
.311- 

V 

154 

45 

0 

785 

i 

=  0.125 

bd 

=  S 

.83o| 

155 

60 

I 

047 

i 

=  0.167 

bd 

=  4 

V 

I56 

75 

I 

309 

A 

=  0.208 

bd 

=  3 

.850^ 

V 

157 

90 

I 

57i 

i 

=  0.250 

bd 

=  3 

•36if 

I58 

H 

105 

I 

833 

& 

=  0.292 

bd 

=  3 

•031- 

159 

H 

120 

2 

094 

i 

=  0.333 

bd 

=  2 

160 

ff 

135 

2 

356 

1 

=  0.375 

bd 

=  2 

.580— 

V 

161 

150 

2 

618 

A 

=  0.417 

bd 

=  2 

.420^ 

162 

165 

2 

880 

H 

=  0.458 

bd 

=  2 

.310^ 

*63 

V 

1  80 

3 

142 

1 

=  0.500 

bd 

—  2 

.200^ 

164 

195 

3 

403 

if 

=  0.541 

bd 

=  2 

.123^ 

165 

V 

2IO 

3 

665 

A 

=  0.583 

bd 

=  2 

.04^ 

166 

£> 

240 

4 

187 

f 

=  0.667 

bd 

=  I 

.931* 

167 

Z/ 

270 

4 

712 

f 

=  0.750 

bd 

=  I 

•85V 

168 

300 

5 

236 

f 

=  01833 

bd 

=  I 

H 

•793- 

169 

102 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  CAST-IRON 
PULLEYS. 

Double  Leather-Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

O 

•524 

*=o 

083 

bd 

=  7 

•?6lf 

170 

45 

O 

.785 

*=0 

125 

bd 

=  5 

H 

•  440- 

171 

60 

I 

.047 

*  =  o 

167 

bd 

=  4 

-< 

172 

75 

I 

.309 

&  =  o 

208 

bd 

=  3 

•59V 

173 

H 

9° 

I 

•571 

i  =  ° 

250 

bd 

=  3 

V 

174 

105 

I 

.833 

ft  =  o 

292 

bd 

=  2 

.827  — 

175 

I2O 

2 

.094 

*«0 

333 

bd 

=  2 

•stff 

I76 

135 

2 

.356 

|  =  o 

375 

bd 

=  2 

H 

•404- 

177 

150 

2 

.618 

A  —  ° 

417 

bd 

=  2 

.*lf 

I78 

I65 

2 

.880 

it  —  ° 

458 

bd 

—  2 

•4 

179 

180 

3 

.142 

|  =  o 

500 

bd 

=  2 

H 

.052  — 

180 

195 

3 

.403 

H  =  o 

541 

bd 

=   I 

.980  — 

z/ 

181 

210 

3 

.665 

&-c 

583 

bd 

=  i 

H 
.909- 

182 

240 

4 

.189 

f±=0 

667 

bd 

=   I 

.804- 
•tf 

183 

27O 

4 

.712 

f  =  0 

750 

bd 

=   1 

.733^ 

184 

300 

5 

.236 

-|  =  o 

833 

bd 

=   I 

•4 

185 

STRENGTH  OF  LEATHER  BELTS. 


103 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  CAST- 
PULLEYS. 

Double  Rawhide- Lacing. 


IRON 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

<tf 

H 

186 

45 

0.785 

i  =  0.125 

bd 

H 
=  5-104- 

187 

60 

1.047 

£  =  0.167 

bd 

H 

1  88 

75 

1.309 

&  =  0.208 

bd 

H 

=  3-372- 

189 

90 

I-57I 

I  =  0.250 

bd 

H 

=  2.943- 

190 

105 

1.833 

•£%  =  0.292 

bd 

=  2.657^ 

191 

120 

2.094 

i  =  0.333 

bd 

-  2.437- 

V 

192 

135 

2.356 

1  =  0.375 

bd 

H 

—  2.225  — 

193 

150 

2.618 

A  =  0.417 

bd 

=  2'Il8f 

194 

i65 

2.880 

M  =  o  458 

bd 

H 
=  2.024  — 

V 

195 

180 

3.142 

i  =  0.500 

bd 

=  1.925  — 

27 

196 

195 

3-403 

tt=  0-541 

bd 

=  I.859f 

197 

2IO 

3-665 

•&  =  0.583 

bd 

=  1.788'!' 

198 

240 

4.189 

£  =  0.667 

bd 

-  r'694f 

199 

270 

4.712 

f  =  0.750 

bd 

=    1.623  — 

z/ 

200 

30O 

5-236 

f  =  0.833 

bd 

=  1.568- 

V 

201 

IO4 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  LEATHER 
PULLEYS. 

Riveted  Joint. 


BELTS  OVER  CAST-IRON 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

b8  = 

5.o6of 

202 

45 

0.785 

i  =  0.125 

b8  = 

3-54sf 

203 

60 

1.047 

i  =  0.167 

b8  = 

H 

2.794- 

2O4 

75 

1.309 

-25¥  =  0.208 

b5  = 

H 

2-343- 

205 

90 

I.57I 

1  =  0.250 

bd  = 

2  .  046  — 

£/ 

206 

105 

1.833 

&  =  °-292 

b8  = 

X.84S? 

207 

120 

2.094 

i  =  0.333 

«r  = 

i.694f 

208 

135 

2.356 

1  =  0.375 

£S  = 

,568f 

209 

150 

2.618 

T*¥  =  0.417 

^o  ^^ 

H 
1-474- 

210 

165 

2.880 

iJ  =  0.458 

3o  ^^ 

H 
1.408- 

211 

180 

3.142 

-£  —  0.500 

^5  = 

H 

1-337- 

212 

195 

3.403 

it  =  0.541 

^5  = 

H 

1-293- 

213 

2IO 

3.665 

A  =  0-583 

b8  = 

1.243- 

214 

240 

4.189 

£  =  0.667 

^5  = 

H 

1.177- 

215 

270 

4.712 

f  =  0.750 

^  = 

I.I2S- 

7^ 

216 

300 

5.236 

f  =  0.833 

b8  = 

H 
i  .  089  — 

z> 

217 

STRENGTH  OF  LEATHER  BELTS.  IO5 

Example.  —  Required  the  width  for  a  leather  belt  -f^ 
inch  thick  which  will  transmit  a  force  of  15  horse- 
power at  a  velocity  of  10  feet  per  second,  the  angle  a 

TT  _  ^  ^ 

being  105°.     In  this  case  —  =    -  =  1.5  and  d  =  ~. 

v        10  16 

Hence  from  formula  (143)  we  have 

*  X  ;=§  =  3-267  g  =  3.267  X  1.5- 

Therefore  b  =  3*267  X  1.5  X  y, 

or,  for  single  leather-lacing, 

b=  15.682"=  15H"  nearly. 
From  formula  (159), 

fX=  3-031  x  1.5, 


16 
=  3.031  X  1.5  X  y, 


or,  for  single  rawhide-lacing, 

b  -  14.549"  = 
From  formula  (175), 


b  X  j-6  =  2.827  X  1.5, 

16 

b  —  2.827  X  1.5  X  •—, 


106  BELTS  AND  PULLEYS. 

or,  for  double  leather-lacing, 

6=  13.570"  =  I3IT- 
From  formula  (191), 

bX  ^6=  2.657  X  1.5, 

16 
*  =  2.657  X  1.5  X  --, 

or,  for  double  rawhide-lacing, 

b  =  12.754"  =  I2f". 
From  formula  (207), 

*X       =  1-848  X  i.S, 


6=  1.848  X  1.5  X      > 


or,  for  a  riveted  joint, 

£    _    O    Q^f^ff      _    QT// 
^7  -    0.070       =    O-^     . 

In  the  majority  of  cases  leather  belts  (single)  are  ap- 
proximately -^  inch  thick.  Very  often  the  arc  em- 
braced by  the  belt  is  180°  —  that  is,  the  pulleys  are 
equal  ;  and,  perhaps,  more  often  the  arc  a  is  about 
135°  =  f  the  circumference.  For  these  cases,  then,  we 
may  obtain  formulas  which  will  prove  very  useful  in 
practice. 


STRENGTH  OF  LEATHER  BELTS.  TO/ 

By  substituting  d  —  -£%  inch  in  formulas  (66),  (82), 
(98),  (114),  (130),  (148),  (164),  (180),  (196),  and  (212), 
successively,  and  reducing,  we  obtain  the  following 
formulas : 

When  a  —  180°  and  6  =  ^  inch, 

Single  leather-lacing,  b  —  O.OI9/P;  ....  (218) 
Single  rawhide-lacing,  b  =  0.0183^;  ....  (219) 
Double  leather-lacing,  £  =  o.oi7iP;  ....  (220) 
Double  rawhide-lacing,  b  —  o.oi6oP',  ....  (221) 
Riveted  joint,  b  =  Q.oinP.  ....  (222) 


TT 

Single  leather-lacing,      b  =  10.839  —  ;      ...     (223) 

TT 

Single  rawhide-lacing,    £=10.057  —  I      •     •     •     (224) 

TT 

Double  leather-lacing,    b—    9.381--;      .     .     .     (225) 

TT 

Double  rawhide-lacing,  b=    8.800  — ;      .     .     .     (226) 

TT 

Riveted  joint  b  =    6.112 — .       .     .     .     (227) 


By  substituting  $  =  -^  inch  in   formulas  (63),  (79), 

(95),  (in)'  (T27)>  (145),  (161),  (177),   (193).  and  (209), 
successively,  we  obtain  the  following  formulas  : 

When  a  =  135°  and  S  —  ^  inch, 

Single  leather-lacing,       b  —  0.023  \P\    ....  (228) 

Single  rawhide-lacing,    b  =  O.O2I4/5;    ....  (229) 

Double  leather-lacing,    b  —  O.O2OO/5;    ....  (230) 

Double  rawhide-lacing,  b  —  O.OiSfP',    ....  (231) 

Riveted  joint,                  £  =  o.oi3oP.     ....  (232) 


108  BELTS  AND  PULLEYS. 

TT 

Single  leather-lacing,      #=  12.699 — ;       .     .     .     (233) 

TT 

Single  rawhide-lacing,    b  =  1 1.794  ;— ;       .     .     .     (234) 

TT 

Double  leather-lacing,    b  =  10.990  -- ;       .     .     .     (235) 

TT 

Double  rawhide-lacing,  b  =  10.171  — ;       ...     (236) 

TT 

Riveted  joint,  b—    7.168—.       .     .     .     (237) 


Example. — Required  the  width  of  a  -^-inch  leather 
belt,  single  leather-lacing,  to  transmit  a  force  of  600 
pounds,  the  pulleys  being  equal. 

From  formula  (218)  we  have 

b  —  0.0197  X  600, 
b=  11.82"  =  I  iff". 

Example.— Required  the  width  of  a  ^--inch  leather 
belt,  single  rawhide-lacing,  to  transmit  a  force  of  15 
horse-power  at  a  velocity  of  10  feet  per  second,  the 
pulleys  being  equal. 

From  formula  (224)  we  have 

b  =  10.057  X  JJ, 


b=  1 5.085" - 
Example. — Required   the  width  of  a  ^--inch  leather 


STRENGTH  OF  LEATHER  BELTS.  1 09 

belt,  double  rawhide-lacing,  to  transmit  a  force  of  600 
pounds,  the   arc  embraced   by  the    belt   being  about 

135°. 

From  formula  (231)  we  have 

b  —  0.0187  X  600, 
b  —  11.22"  =  1 1 3V'. 

Example. — Required  the  width  of  a  ^--inch  leather 
belt,  riveted  joint,  to  transmit  a  force  of  15  horse-power 
at  a  velocity  of  10  feet  per  second,  the  arc  embraced 
by  the  belt  being  about  135°. 

From  formula  (237)  we  have 


=  7.168  X  — , 
10 


b  =  10.75"  =  lof ". 

,    The  following  tables  give  the  forces  in  pounds  (/>), 
and   the  values   of  the   horse -power  divided    by   the 


velocity  in  feet  per  second  I  — 1,  corresponding  to  dif- 
ferent widths  of  ^--inch  leather  belts  from  I  inch  to  30 
inches  for  a  —  180°  and  a  =  135°  for  each  of  the  five 
methods  of  joint-fastening  given  above.  For  a  great 
many  cases  which  arise  in  practice  the  tables  will  prove 
convenient  and  labor-saving. 


1 10 


BELTS  AND  PULLEYS. 


TABLE  OF  WIDTHS  OF  LEATHER  BELTS  OVER  CAST-IRON  PULLEYS, 
WHEN  a  —  1 80°  AND  d  =  ¥y.     From  Formulas  (2iS)-(222). 


Width 
in 
inches. 

/>,  single 
leather- 
lacing. 

P,  single 
rawhide- 
lacing. 

/>,  double 
leather- 
lacing. 

P,  double 
rawhide- 
lacing. 

P,  riveted 

joints. 

No. 

I 

50.76 

54.64 

58.48 

62.50 

90.09 

I 

Ii 

76.14 

81.97 

87.72 

93-75 

135.14 

2 

2 

101.52 

109.29 

116.96 

125.00 

180.18 

3 

2| 

126.90 

136.61 

146.20 

156.25 

225.23 

4 

3 

152.28 

163.93 

175-44 

187-50 

270.27 

5 

3i 

177-66 

191  .26 

204.68 

218.75 

315.32 

6 

4 

203.05 

218.58 

233.92 

250.00 

360  .  36 

7 

4i 

228.43 

245.90 

263.16 

281.25  . 

405.41 

8 

5 

252.71 

2/3.22 

292.40 

312.50 

450.45 

9 

51 

279.19 

300  .  46 

321   64 

343-75 

495.50 

10 

6 

304.57 

327.87 

350.88 

375-00 

540.54 

ii 

7 

355-33 

382.51 

409.36 

437-50 

630.63 

12 

8 

406  .  09 

437.16 

467.84 

500.00 

720.72 

13 

9 

456.85 

491  .80 

526.32 

562.50 

810.81 

14 

10 

507-61 

546.45 

584.80 

625.00 

900  .  90 

15 

ii 

558.38 

601.09 

643.27 

687.50 

990.99 

16 

12 

608.57 

655.74 

701.75 

750.oo 

i  08  i  .08 

17 

14 

710.00 

765.03 

818.71 

875.00 

1261.26 

18 

16 

812.18 

874.32 

935.67 

1000.00 

1441.44 

19 

18 

9I3.7I 

983  61 

1052.63 

1125.00 

1621.62 

20 

20 

1015.22 

1092  .  90 

1169.59 

1250.00 

1801.80 

21 

22 

1116.75 

1202.19 

1286.55 

1375.00 

1981.98 

22 

24 

1218.27 

1311.48 

1403.51 

1500.00 

2162.16 

23 

26 

1319.79 

1400.77 

1520.47 

1625.00 

2342.34 

24 

28 

1421.31 

1530.05 

1637.43 

1750.00 

2522.52 

25 

30 

1522.84 

I639-34 

1754.44 

1875.00 

2702.70 

26 

STRENGTH  OF  LEATHER  BELTS. 


II] 


TABLE  OF  WIDTHS  OF   LEATHER  BELTS  OVER  CAST-IRON  PULLEYS, 
WHEN  a  =  180°  AND  £  =  -£%" .     From  Formulas  (223)-(227). 


Width 
in 
inches. 

H 
~  ,  single 

leather- 
lacing. 

H 

—  ,  single 

rawhide- 
lacing. 

H 

—  ,  double 

leather- 
lacing. 

H 

—  ,  double 

V 

rawhide- 
lacing. 

H 
—  ,  riveted 

joints. 

No. 

I 

0.0923 

0.0994 

O.IO66 

0.1136 

0.1636 

I 

Ii 

0.1384 

0.1491 

0.1599 

0.1705 

0.2454 

2 

2 

0.1845 

0.1989 

0.2132 

0.2273 

0.3272 

3 

2i 

0.2307 

0.2486 

0.2665 

0.2841 

O  .  4090 

4 

3 

0.2768 

0.2983 

0.3198 

0.3409 

0.4908 

5 

3* 

0.3229 

0.3480 

0.3731 

0.3977 

0.5726 

6 

4 

0.3690 

0.3977 

0.4264 

0.4546 

0.6544 

7 

4* 

0.4152 

0.4474 

o  4797 

0.5II4 

0.7362 

8 

5 

0.4613 

0.4972 

0-5330 

0.5682 

0.8181 

9 

5* 

0.5074 

0.5469 

0.5863 

0.6250 

0.8999 

10 

6 

0.5536 

0.5966 

0.6396 

0.6818 

0.9817 

ii 

7 

0.6458 

o  .  6960 

0.7462 

0-7955 

I.I453 

12 

8 

0.7381 

0-7954 

0.8528 

0.9091 

1.3089 

13 

9 

0.8303 

o  .  8949 

0-9594 

1.0228 

1.4725 

14 

10 

0.9226 

0.9943 

1.0659 

1.1364 

1.6361 

15 

ii 

.0149 

1.0937 

1.1726 

1.2500 

i  •  7997 

16 

12 

.1071 

1.1932 

1.2792 

1.3637 

1-9633 

17 

14 

.2916 

1.3920 

1.4924 

1.5910 

2  .  2905 

18 

16 

.4762 

1.5909 

1.7056 

1.8182 

2.6178 

19 

18 

.6607 

1.7897 

1.9188 

2-0455 

2.9450 

20 

20 

.8452 

1.9886 

2.1318 

2.2728 

3.2722 

21 

22 

2.0297 

2.1875 

2.3452 

2.5001 

3-5994 

22 

24 

2.2142 

2.3863 

2.5584 

2.7274 

3.9266 

23 

26 

2.3988 

2.5852 

2.7716 

2.9546 

4.2539 

24 

28 

2.5833 

2.7840 

2.9848 

3-1819 

4.58H 

25 

30 

2.7678 

2.9829 

3.1977 

3.4092 

4.9083 

26 

112 


BELTS  AND  PULLEYS. 


TABLE  OF  WIDTHS  OF   LEATHER  BELTS  OVER  CAST-IRON  PULLEYS, 
WHEN  a  =  135°  AND  d  =  -/%".     From  Formulas  (228)-(232). 


Width 
in 
inches. 

P,  single 
leather- 
lacing. 

P,  single 
rawhide- 
lacing. 

P.  double 
leather- 
lacing. 

/»,  double 
rawhide- 
lacing. 

/>,  riveted 
joints. 

No. 

I 

43.29 

46.73 

50.00 

53.48 

76.92 

I 

I* 

64.94 

70.09 

75.00 

80.21 

115.38 

2 

2 

86.58 

93.46 

IOO.OO 

106.95 

153.85 

3 

2* 

108.23 

116.82 

125.00 

133.69 

192.31 

4 

3 

129.87 

140.19 

150.00 

160.43 

230.77 

5 

3* 

151.51 

163.55 

175.00 

187.17 

269.23 

6 

4 

173.16 

186.92 

200.00 

213.90 

307.69 

7 

44 

194.81 

210.28 

225.00 

240.64 

346.15 

8 

5 

216.45 

233.65 

250.00 

267.38 

384.62 

9 

54 

238.10 

257-01 

275.00 

294  .  I  2 

423.08 

10 

6 

259.74 

280.37 

300.00 

320.86 

461.54 

ii 

7 

303.03 

327.10 

350.00 

374-33 

538.46 

12 

8 

346.32 

373.83 

400  .  oo 

427.81 

615-38 

13 

9 

389.61 

420.56 

450.00 

481.28 

692.31 

14 

10 

432.90 

467.29 

500  .  oo 

534.76 

769.23 

15 

ii 

476.19 

514.02 

550.00 

588.24 

846.15 

16 

12 

519.48 

560.75 

600.00 

641.71 

923.08 

17 

14 

606  .  06 

654-21 

700.00 

748.66 

1076.92 

18 

16 

692  .  64 

747.66 

800.00 

845.62 

1230.77 

19 

18 

779-22 

841.12 

900.00 

962.57 

1384.61 

20 

20 

865.80 

934.58 

1000.00 

1069.52 

1538.46 

21 

22 

952.38 

1028.04 

1100.00 

1176.47 

1692.31 

22 

24 

1038.96 

II2I.5O 

I  200.00 

1283.42 

1846.15 

23 

26 

H25.54 

1214.95 

1300.00 

1380.38 

2000  .  00 

24 

28 

1212.12 

1308.41 

1400.00 

1497.33 

2153.84 

25 

30 

1298.70 

1401.87 

1500.00 

1604.28 

2307.69 

26 

STRENGTH    OF  LEATHER  BELTS. 


TABLE  OF  WIDTHS  OF  LEATHER  BELTS  OVER   CAST-IRON   PULLEYS, 
WHEN  a  =  135°  AND  d  =  37¥".     From  Formulas  (233H.237). 


Width 
ia 
inches. 

—  ,  single 

leather- 
lacing. 

£  single 

rawhide- 
lacing. 

—  ,  double 

V 

leather- 
lacing. 

—  ,  double 

V 

rawhide- 
acing. 

—  ,  riveted 

V 

joints. 

No. 

I 

0.0787 

0.0879 

0.0910 

0.0983 

0.1395 

I 

i* 

o.  1181 

0.1272 

0.1365 

0.1475 

0.2093 

2 

2 

0.1575 

0.1696 

0.1820 

o.  1966 

0.2790 

3 

2£ 

o.  1969 

O.2I2O 

0.2275 

0.2458 

0.3488 

4 

3 

0.2362 

0.2544 

0.2730 

0.2950 

0.4185 

5 

3i 

0.2756 

0.2968 

0.3185 

0.3441 

0.4883 

6 

4 

0.3150 

0.3392 

0.3640 

0-3933 

0.5580 

7 

4* 

0-3544 

o  3816 

0.4095 

0.4424 

0.6278 

8 

5 

0.3937 

0.4239 

0.4550 

0.4916 

0.6976 

9 

5^ 

0.4331 

0.4663 

0.5004 

0.5108 

0.7673 

10 

6 

0.4725 

0.5087 

0-5459 

0.5899 

0.8371 

ii 

7 

0.5512 

0.5935 

0.6396 

0.6882 

0.9766 

12 

8 

0.6300 

0.6783 

0.7279 

0.7866 

1.1161 

13 

9 

0.7087 

o  7631 

0.8189 

0.8849 

1.2556 

14 

10 

0.78/5 

0.8479 

o  .  9099 

0.9832 

I.395I 

15 

ii 

0.8662 

0.9327 

J  .  OOO9 

.0815 

1.5346 

16 

12 

0.9450 

.0175 

1.0919 

.1799 

1.6741 

17 

14 

.  1024 

.1870 

1-2739 

•3765 

1-9532 

18 

16 

.2600 

.3566 

1.4558 

•5731 

2.2322 

19 

18 

.4174 

.5262 

1.6378 

.7698 

2.5112 

20 

20 

•  5749 

.6958 

1.8198 

1.9664 

2.7902 

21 

22 

.7324 

1.8654 

2.0018 

2.  1630 

3.0692 

22 

24 

.8900 

2.0349 

2.1838 

2-3597 

3.3482 

23 

'    26 

2.0475 

2  .  2045 

2.3057 

2.5563 

3.6273 

24 

28 

2  .  2048 

2.3741 

2.5477 

2.7530 

3.9063 

25 

30 

2.3623 

2.9676 

2.7297 

2.9496 

4-1853 

26 

Example. — Required  the  force  in  pounds  which  can 
be  safely  transmitted  by  a  leather  belt  20  inches  wide 
and  3^  inch  thick,  running  over  two  pulleys  of  equal 
diameters  (a  =  180°),  the  joint  being  fastened  by  a 
double  rawhide-lacing. 

In  the  table  on  page  no,  column  of  belt-widths,  line 
21,  we  find  our  width  of  20  inches,  and  corresponding 


114  BELTS  AND  PULLEYS. 

to  this,  in  the  column  for  double  rawhide-lacing,  we 
find  the  required  force  P  =  1250  pounds. 

Example. — Required  the  width  of  a  leather  belt  -^ 
inch  thick,  which  will  safely  transmit  a  force  of  1000 
pounds  running  over  two  pulleys  of  equal  diameters, 
the  fastening  being  a  riveted  joint. 

In  the  table  on  page  no,  column  for  riveted  joints, 
line  17,  we  find  P  =  1081.08  pounds, — the  nearest 
value,  not  less  than  1000  pounds, — and,  in  the  column 
for  belt-widths,  we  find  the  value  corresponding  to 
P=  1081.08,  b  =  12  inches. 

Example. — Required  the  horse-power  which  can  be 
safely  transmitted  by  a  leather  belt  12  inches  wide  and 
-£%  inch  thick,  running  over  two  pulleys  of  equal  diame- 
ters at  a  velocity  of  15  feet  per  second,  the  fastening 
being  a  single  rawhide-lacing. 

In  the  table  on  page  ill,  column  of  belt-widths,  line 
17,  we  have  b  =  12  inches,  and,  in  the  column  for  single 
rawhide-lacing,  the  corresponding  value 

H 

-  —  1.1932. 

V 

TT 

Hence      —  =  1.1932,        H  —  15  X  I.I932> 

or  H—  17.90. 

Example. — Required  the  velocity  at  which  a  leather 
belt  12  inches  wide  and  ^  inch  thick  can  be  driven 
over  two  pulleys  of  equal  diameters,  in  order  to  transmit 
a  force  of  17.90  horse-power,  the  fastening  being  a  sin- 
gle rawhide-lacing. 


LEATHER-COVERED  PULLEYS.  115 

In  the  table  on  page  in,  column  for  single  rawhide- 
lacing,  we  find,  corresponding  to  a  belt-width  of  12 
inches, 

H 

-  =  1. 1032. 

V 

17.90  17.90 

Consequently  -       -  =  1.1932,         v  =  -      — , 
v  1.1932 

or  v  —  15  feet  per  second. 

Example. — Given  the  data  a  =  135°,  d  =  •£%•  inch, 
H  ~  30,  v  =  15,  double  leather-lacing,  required  the 
belt-width.  In  this  case 

£=30  = 

v        15 

The  table  on  page  113,  column  for  double  leather-lac- 
ing, line  22,  gives 

H 

-  —  2.0018, 
v 

and  a  corresponding  belt-width  of  b  =  22  inches. 

§  ii.     Leather  Belts  over  Leather-covered  Pulleys. 

As  we  have  demonstrated  in  the  foregoing  pages, 
the  average  leather  belt  will  not  transmit  a  force  equal 
to  its  strength,  for  the  reason  that  it  will  slip  upon  its 
pulley  before  it  will  break.  If  we  can  conveniently  in- 
crease the  adhesion  between  the  belt  and  pulley, — i.e., 
increase  the  coefficient  of  friction, — and  in  this  way  pre- 


Il6  BELTS  AND  PULLEYS. 

vent  slipping,  the  belt  can  be  made  to  do  more  work 
without  increasing  its  size.  Various  methods  have 
been  from  time  to  time  proposed  for  obtaining  a  greater 
coefficient  of  friction,  such  as  coating  the  pulley-faces 
with  gum,  rosin,  etc.;  but  these  methods  have  more 
often  than  otherwise  proved  useless,  from  the  fact  that 
the  belt  is  soon  rendered  stiff  and  clumsy  by  the  sub- 
stance placed  upon  the  face  of  the  pulley.  Probably 
the  best  of  all  contrivances  in  use  for  this  purpose  is 
the  pulley  with  a  leather-covered  face.  The  leather  is 
easily  fastened  securely  upon  the  pulley,  and  we  have 
then  practically  a  leather  belt  running  over  a  leather 
pulley.  A  series  of  carefully  tried  experiments  has 
given  the  coefficient  of  friction  for  leather  belts  over 
leather-covered  pulleys  equal  to  0.45-0.05  greater  than 
that  for  leather  belts  over  cast-iron  pulleys.* 

If  we  substitute  q>  =  0.45  successively  in  formulas 
(40),  (41),  and  (42),  and  reduce,  we  shall  obtain  for 
leather  belts  over  leather-covered  pulleys  the  follow- 
ing expressions : 

T 

logy   =  0.1953^;         ....       (238) 

when  a  is  expressed  in  circular  measure, 

T 
log--  =  0.00341^;    ....     (239) 

T 
*  Reuleaux  says  :   "For  a  covering  entirely  new  the  value  of  —  is 

between  6  and  7;  after  some  service  this  value  decreases,  but  still  does 
not  become  less  than  4  to  5 ;  the  arc  embraced  by  the  belt  being  equal 

T 
to  it.     The  smaller  value,  i.e.,  —  =  4,  corresponds  to  0.44  for  the 

coefficient  of  friction."     See  also  Appendix  I. 


LEA  THE  2^-  CO  VERED  P  ULLE  VS. 

when  a  is  expressed  in  degrees, 


log--  =  1.2280:; 


117 


(240) 


when  a  is  expressed  in  fractions  of  the  circumference. 
The    following    table,    calculated    from    the    above 

T 

formulas,  gives  values  of  —    for  different  values    of  a 

from  30°  to  300°.      The  arrangement  is  similar  to  that 
of  the  table  on  page  88. 

TABLE  OF  TENSIONS  FOR  LEATHER  BELTS  OVER   LEATHER-COVERED 

PULLEYS. 


In  degrees. 

In  circular 
measure. 

In  fractions  of 
circumference. 

t 

50 

0.524 

^  =  0.083 

1.266 

45 

0.785 

i  =  0.125 

1.424 

60 

1.047 

£  =  0.167 

1.601 

75 

1.309 

•jfe  =  0.208 

1.802 

90 

I-57I 

i  =  0.250 

2.027 

105 

1.833 

A  =  °-292 

2.281 

120 

2.004 

i  =  0.333 

2.566 

IS5 

2.356 

1  =  o.375 

2.886 

150 

2.618 

A  =  0-4i7 

3-247 

165 

2.880 

H  =  0.458 

3-653 

180 

3.142 

J  =  0.500 

4.110 

193 

3.403 

if  =  0.541 

4-623 

210 

3.665 

A  =  0.583 

5.201 

240 

4.189 

t  =  0.667 

6.583 

270 

4.712 

f  =  0.750 

8.331 

3OO 

5-236 

1-0.833 

12.655 

T 


By  substituting  the  successive  values  of  -—  from  the 

above  table  in  formula  (48),  we  obtain  the  following 
table,  similar  to  the  one  on  page  90 : 


US 


BELTS  AND  PULLEYS. 


TABLE  OF  GREATEST  TENSION  FOR  LEATHER  BELTS  OVER  LEATHER- 
COVERED  PULLEYS. 


a  = 

T=PX 

In  degrees. 

In  circular 
measure. 

In  fractions  of 
circumference. 

30 

0.524 

rV  =  0.083 

4.76 

45 

0.785 

i  -  0.125 

3.36 

60 

1.047 

i  =  0.167 

2.66 

75 

1.309 

^  =  0.208 

2.25 

90 

I.57I 

J  =  0.250 

1.97 

105 

1.833 

^  =  0.292 

1.79 

120 

2.094 

i  =  0.333 

1.64 

135 

2.356 

1-0.375. 

1-53 

150 

2.618 

•h  —  0.417 

1.44 

I65 

2.880 

it  -0.458 

1.38 

1  80 

3.142 

i  =  0.500 

1.32 

IQ5 

3.403 

if  —  0.541 

1.28 

210 

3.665 

A  -0.583 

1.24 

240 

4.189 

f  =  0.667 

Z.ll 

270 

4.712 

f  =  0.750 

1.14 

300 

5-236 

1-0.833 

1.09 

Example. — A  leather  belt  running  over  a  leather- 
covered  pulley  transmits  a  force  of  500  pounds.  It  is 
required  to  determine  the  greatest  tension  on  the  belt, 
assuming  that  the  belt  embraces  f  the  circumference 
of  the  pulley.  From  the  table  we  find,  corresponding 
to  a  =  •§  the  circumference, 

T  =  P  X  1.18  =  500  X  1.18, 
or  T  —  590  pounds. 

Example. — The  greatest  tension  on  a  leather  belt, 
running  over  a  leather-covered  pulley  and  embracing  £ 
the  circumference,  is  T  —  792  pounds.  Required  the 
force  in  pounds  which  it  can  transmit.  The  table  gives 

T=PX  1.32, 


LEATHER-COVERED   PULLEYS. 

as  the  greatest  tension  corresponding  to  a  = 
cumference.     Hence 


the  cir- 


or 


792  ==  P  X  1.32, 
P  =  600  pounds. 


792. 
1.32' 


By  substituting  the  values  of  T'from  the  above  table 
successively  in  formulas  (51),  (52),  (53),  (54),  and  (55), 
the  following  tables  of  formulas  have  been  obtained. 

The  application  of  these  formulas  will  be  easily  un- 
derstood from  the  explanation  of  the  similar  tables  on 
pages  96-98. 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER- COVERED 

PULLEYS. 

Single  Leather-Lacing. 


a.  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A  =  0-083 

bd 

=  O.OI464/7 

241 

45 

0.785 

\  =  0.125 

bd 

=  O.OI034/* 

242 

60 

1.047 

i  =  0.167 

tf 

=  o.ooSiS/3 

2^3 

75 

1.309 

-£%  =  0.208 

bd 

=  0.00692^ 

244 

90 

I-57I 

{  =  0.250 

bd 

=  o.  00606  P 

245 

105 

1.833 

A  =  0-292 

bd 

—  0.005  5  1/* 

246 

120 

2.094 

i  =  0.333 

bd 

—  o.  00503  P 

247 

135 

2.356 

-1  =  0.375 

bd 

—  0.0047I/7 

248 

150 

2.618 

T2    =   0.417 

bd 

=  o.  00443  P 

249 

I65 

2.880 

tt  =  0.458 

bd 

—  0.00424/5 

250 

180 

3.142 

4-  =  0.500 

bd 

=  o.  00406  P 

251 

195 

3.403 

if  =  0.541 

bd 

—  o.  00394^ 

252 

2IO 

3-665 

A  =  0.583 

bd 

—  0.003827' 

253 

240 

4.189 

*  =  0.667 

bd 

=  0.00363^ 

254 

270 

4.712 

f  =  0.750 

bd 

=  0.00351^ 

255 

300 

5-236 

1  =  0.833 

bd 

=  0.00335^ 

256 

I2O 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER-COVERED 

PULLEYS. 
Single  Rawhide- Lacing. 


o.  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

£5 

=  0.  01  360/5 

257 

45 

0.785 

i  =  0.125 

M 

=  0.009607* 

258 

60 

1.047 

|  =  0.l67 

bd 

=  0.007607* 

259 

75 

1.309 

•^  =  0.208 

t>8 

—  o.  00643  P 

260 

90 

I-57I 

£  :      0.250 

is 

—  0.005637* 

261 

105 

.1.833 

¥7T  =.  0.292 

ltd 

=  0.005  1  iP 

262 

120 

2.094 

£  —  0.333 

bd 

=  0.00469^ 

263 

135 

2.356 

I  =  0.375 

bd 

—  0.004377^ 

264 

150 

2.618 

T%  =  0-4I7 

bd 

—  0.0041  iP 

265 

I65 

2.880 

H-  =  0.458 

bd 

—  0.00394/3 

266 

180 

3.142 

•j-  =  0.500 

bd 

=  0.003777^ 

267 

195 

3.403 

M  =  0.541 

bd 

=  0.003667-' 

268 

210 

3.665 

A  =  °-5S3 

bd 

—  0.003547* 

269 

240 

4.189 

\  =  0.667 

bd 

=  0.0033  7  P 

270 

270 

4.712 

f  =  0.750 

bd 

—  0.003267* 

271 

3OO 

5-236 

\  =  0.833 

bd 

=  0.003117* 

272 

TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER-COVERED 

PULLEYS. 
Double  Leather-Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

bd  =  0.012697* 

273 

45 

0.785 

i  =  0.125 

£<5  =  0.008967* 

274 

60 

1.047 

•J-  -  0.167 

bd  =  0.007097* 

275 

75 

1.309 

-f%  =  0.208 

^<5  —  o  00600  P 

276 

90 

I.57I 

J  =  0.250 

bd  =  0.005257* 

277 

105 

1-833 

A  =  0-292 

bd  =  0.004777* 

278 

I2O 

2.094 

i  =  0.333 

M  =  0.004377* 

279 

135 

2.356 

1  =  0.375 

bd  =  0.004087* 

280 

150 

2.618 

A  =  0.417 

/;<5  =  0.003847* 

281 

I65 

2.880 

tt  =  °-458 

bd  =  0.003687* 

282 

ISO 

3.142 

-£-  —   O.5OO 

bd  =  0.003527* 

283 

195 

3-403 

if  =  0.541 

/^5  =  0.003417* 

284 

210 

3.665 

A  =  °-583 

bd  =  0.003317* 

285 

240 

4.189 

f^o.667 

/65  =  0.003157* 

286 

270 

4.712 

I  =  0.750 

bd  =1  0.003047* 

287 

300 

5-236 

1  =  0.833 

bd  =  0.0029  1  7* 

288 

LEATHER-COVERED  PULLEYS.  121 

^^ 

TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER-COVERED 

PULLEYS. 

Double  Raw  hide- Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a.  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

bd 

=  o.ongoP 

289 

45 

0.785 

i  =  0.125 

bd 

=  o.oo$4oP 

290 

60 

1.047 

i  =  0.167 

M 

=  0.00665^ 

291 

75 

1.309 

-£%  =  0.208 

M 

=  O.OO563/3 

292 

90 

I-57I 

J  =  0.250 

£5 

=  0.00493/5 

293 

105 

1.833 

£  =  0.292 

** 

=  o.  00448^ 

294 

I2O 

2.094 

t  =  0-333 

M 

=  O.OO4IO/* 

295 

135 

2.356 

-1  =  0.375 

M 

=  o.  00383^ 

296 

150 

2.618 

T5¥  =  0.417 

M 

=  0.00360/* 

297 

165 

2.880 

ii  =  0.458 

£S 

—  O.OO345/' 

298 

1  80 

3.142 

-J-  =  o  .  500 

bd 

=  0.00330/3 

299 

195 

3.403 

M  =  0.541 

bd 

=  O.OO32O/* 

300 

2IO 

3-665 

A  =  0.583 

bd 

=  O.OO3IO/* 

301 

240 

4.189 

|  =  0.667 

bd 

—  o.  00295  P 

302 

270 

4.712 

f  =  0.750 

bd 

=  0.09285/5 

303 

300 

5.236 

I  =  0.833 

bd 

=  O.O0273/* 

304 

TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER-COVERED 

PULLEYS. 
Riveted  Joint. 


a.  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A  =  0.083 

bS 

=  0.00828/5 

305 

45 

0.785 

i  =  0.125 

bS 

=  0.005847' 

306 

60 

1.047 

i  =  0.167 

bS 

=  o.  00463  P 

307 

75 

1.309 

^  =  0.208 

dd 

=  o.oo39i/> 

308 

90 

I.57I 

i  =  0.250 

b$ 

=  0.00343^ 

309 

105 

1.833 

^  —  0.291 

1>S 

=  0.003  u  /* 

310 

I2O 

2.094 

i  =  o-333 

it 

—  o.  00285  P 

311 

135 

2.356 

f  =  0.375 

b8 

=  o.  00266  P 

312 

150 

2.618 

•fs  =  0.417 

M 

=  0.002507* 

313 

I65 

2.880 

tt-  0.458 

bd 

=  o.  00240  P 

3H 

180 

3.142 

\  =  0.500 

b8 

=  0.0022qP 

315 

J95 

3.403 

if  =  0.541 

b8 

=  0.00222P 

316 

210 

3-665 

A  =  0-583 

bd 

=  O.OO2I6P 

317 

240 

4.189 

1  =  0.667 

bS 

=  o.  00205  P 

318 

270 

4.712 

f  =  0.750 

bd 

—  o.ooigSP 

319 

300 

5.236 

1  =  0.833 

bd 

=  o.ooigoP 

32O 

122  BELTS  AND  PULLEYS. 

Example. — A  leather  belt  J  inch  thick,  running  over  a 
leather-covered  pulley,  transmits  a  force  of  500  pounds. 
Required  the  width  of  the  belt  for  single  leather-lacing 
and  single  rawhide-lacing,  taking  a  =  45°.  From 
formula  (242)  we  have 

/;  X  i  =  0.01034  X  500,     b  =  0.01034  X  500  X  4, 
or,  for  single  leather-lacing, 

b  =  20.68"  =  20^-"  nearly. 
From  formula  (258)  we  have 

b  X  i  =  0.00960  X  500,     b  —  0.00960  X  500  X  4, 
or,  for  single  rawhide-lacing, 

b  =  19.20"  =  I9|f.* 

Example. — With  the  data  a=  1.833,  circular  meas- 
ure, d  =  %  inch,  and  b  —  20  inches,  required  the  forces 
in  pounds  which  the  belt  can  transmit  for  each  of  the 

*  If  we  take  the  above  data,  P  =  500,  a  —  45°,  d  —  \  inch,  and  cal- 
culate the  width  of  a  leather  belt  running  over  a  cast-iron  pulley,  we 
shall  have,  from  formula  (57),  for  single  leather-lacing,  b  —  0.01142 
X  500  X  4  =  22.84  inches.  The  difference  between  the  widths  of  the 
belt  necessary  for  transmission  over  cast-iron  and  leather-covered  pul- 
leys is  therefore  22. 84  —  20.68  =  2.16  inches,  which  shows  a  gain  for  the 
leather-covered  pulley  of  nearly  10  per  cent  over  the  cast-iron  pulley. 


LEATHER-COVERED  PULLEYS.  12$ 

above  methods  of  joint-fastening,  supposing  the  belt 
to  run  over  a  leather  covered  pulley. 
From  formula  (246)  we  have 


20  X  0.25 
20  X±  =  0.0055  iP,         P--    -- 


or,  for  single  leather-lacing, 

P  —  907.44  pounds. 
From  formula  (262), 

20  X  0.25 

20  X  J  —  0.005  I  IP,  P  — 

or,  for  single  rawhide-lacing, 

P  —  978.47  pounds. 
From  formula  (278), 


or,  for  double  leather-lacing, 

/*=  1048.22  pounds. 


124  BELTS  AND  PULLEYS. 

From  formula  (294), 


2O  X  O 

X  i  =  o. 


or,  for  double  rawhide-lacing, 

/>  =  1116.07  pounds. 
From  formula  (310), 


or,  for  a  riveted  joint, 

P=  1607.71  pounds. 
The   formulas  of  the  following  tables,  obtained  by 

TT 

substituting  P=  550  —  in  formulas  (241)  to  (320),  and 

similar  to  the  formulas  on  pages  100-104,  will  prove 
convenient  in  calculating  widths  of  leather  belts  over 
leather-covered  pulleys  from  the  horse-power  trans- 
mitted and  the  velocity  in  feet  per  second  : 


LEA  7^HER-CO  VERED  P ULLE  YS. 


125 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER  COVERED 

PULLEYS. 

Single  Leather  Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A 

=   0.083 

bd 

=  8 

H 

V 

321 

H 

45 

0.785 

4 

=   0.125 

bd 

=  5 

687- 

322 

V 

60 

1.047 

1 
6" 

=   0.167 

bd 

=  4 

H 

499- 

323 

H 

75 

1.309 

* 

=   0.208 

bd 

—  3 

806- 

V 

324 

H 

90 

I-57I 

i 

=   0.250 

bd 

=  3 

333- 

325 

105 

1-833 

A 

=   0.292 

bd 

=  3 

95«f 

326 

120 

2.094 

i 

=   0-333 

bd 

=  2 

.767f 

327 

135 

2.356 

f 

=   0-375 

bd 

=  2 

H 

•59!- 

328 

H 

150 

2.618 

A 

=   0.417 

bd 

=  2 

•437- 

329 

I65 

2.880 

H 

=   0.458 

bd 

=  2 

H 

'332~ 

330 

180 

3.142 

* 

=   0.500 

bd 

=  2 

H 
•233- 

331 

195 

3.403 

« 

=   0.541 

bd 

=  2 

167- 

332 

V 

210 

3.665 

_7_ 

=0.583 

bd 

=  2 

H 

.  IOI  — 

333 

240 

4.189 

f 

=   0.667 

bd 

=  , 

H 

•997- 

334 

27O 

4.712 

f 

=   0.750 

bd 

=   I 

H 

•93i- 

335 

H 

300 

5-236 

f 

=0.833 

bd 

=  I 

843f 

336 

126 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS 


FOR  LEATHER  BELTS  OVER  LEATHER-I 
PULLEYS. 

Single  Raw  hide -Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0.083 

TT 

bd  =  7.480  — 

337 

45 

0.785 

t  =  °-I25 

bd  =  5-280^ 

338 

60 

1.047 

i  =  0.167 

TT 

bd  =  4.180  — 

339 

75 

1.309 

-fz  —  0.208 

b$  =  3.537- 

340 

90 

I-57I 

i  =  0.250 

68  =  3-097- 

341 

105 

1.833 

A  =  0.292 

bd  =  2.8ll~ 

342 

120 

2.094 

i  =  0-333 

bd  =  2.580^ 

V 

343 

135 

2.356 

*  -0.375 

LX                   H 

bo  =  2.404  — 

V 

344 

150 

2.618 

T¥  =  0.4I7 

T  r 

b$  =  2.261  — 

V 

345 

I65 

2.880 

tt  =  0-458 

TT 

bd  =  2.167  — 

346 

180 

3.142 

-J-  =  0.500 

TT 

bd  =  2.074  — 

'^  V 

347 

195 

3-403 

if  =  0.541 

bd  =  2.013  — 

V 

348 

210 

3.665 

A  =  0.583 

bd  =  1-947- 

349 

24O 

4.189 

t  =  0.667 

bd  =  1.854^ 

350 

270 

4.712 

l  =  0.750 

bd  =  i  .  793  - 

35i 

3OO 

5.236 

1  =  0.833 

H 

&o  =  i.  701  — 
f 

352 

LEA  THER-  CO  VERED  P  ULLE  YS. 


127 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER  COVERED 

PULLEYS. 

Double  Leather-Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

•fg  =  0.083 

TT 

t>5  =  6.980— 
t» 

353 

45 

0.785 

i  =  0.125 

^ 

dS  =  4.928  — 
^  v      z/ 

354 

60 

1.047 

i  =  0.167 

H 

bS  =  3.900  — 

355 

75 

1.309 

^  =  0.208 

H 

b§  =  3  .  300  — 
J  J      v 

356 

90 

I-57I 

£  =  0.250 

H 

35  =  2.888  — 

357 

105 

1.833 

^  —  0.292 

H 

b§  —  2.624- 

'  z/ 

358 

120 

2.094 

i  =  0.333 

^ 

^5  =  2.404- 

359 

135 

2.356 

1  =  0.375 

^ 

M  -2.244- 

360 

150 

2.618 

•fs  =  0.417 

II 

b§  —  2.II2  — 

361 

165 

2.880 

«  =  0.458 

H 
b8  =  2.024  — 
^» 

362 

180 

3.142 

\  =  0.500 

TT 

bS=  1.936- 

363 

195 

3.403 

H  =  °-54i 

^5  -  1.876^ 

364 

210 

3.665 

A  =  0.583 

z> 

365 

240 

4.189 

f  =  0.667 

^  =  1.733- 

366 

270 

4.712 

$  =  0.750 

/^5  —  i  .672  — 

367 

300 

5-236 

f  -0.833 

35  —  1.601  — 

z/ 

368 

128 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  LEATHER-BELTS  OVER  LEATHER-COVERED 

PULLEYS. 

Double  Rawhide-Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

ff 

30 

0.524 

•A-  =  °  °83 

bd  =  6.545- 

369 

H 

45 

0.785 

I  =  0.125 

bd  =  4.620  — 

370 

H 

60 

1.047 

i  =  0.167 

371 

ff 

75 

1.309 

-ff  =  0.208 

bd  -.  3-097- 

372 

ff 

90 

T-57I 

i  =  0.250 

bd  =  2.712  — 

373 

H 

105 

1-833 

^  =  0.292 

**  =  2.464- 

374 

H 

120 

2.094 

i  =  0.333 

bd  =  2,255— 

375 

// 

135 

2.356 

1  =  0.375 

bd  =  2.107  — 

'  V 

3/6 

H 

150 

2.6I8 

A-  =  0-417 

bd  =  1.980- 

V 

377 

77 

165 

2.880 

•jj  —  0.458 

bd  =  1.898- 

378 

» 

H 

1  80 

3.142 

i  =  0.500 

bd  =  1.815^- 

379 

H 

195 

3.403 

tt  =  0.541 

bd  =  i  .760  — 

380 

H 

210 

3.665 

ft-  =0.5*3 

bd  =  1.705- 

38i 

^7 

240 

4.189 

f  =  0.667 

^  =  1.623- 

382 

^ 

27O 

4.712 

f  =  0.750 

££  =  1.568  — 

0      z/ 

383 

H 

3OO 

5.236 

1  =  0.833 

bd  =  1.502  — 

384 

LEATHER-COVERED  PULLEYS. 


129 


TABLE  OF  FORMULAS  FOR  LEATHER  BELTS  OVER  LEATHER-COVERED 

PULLEYS. 

Riveted  Joints. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A  =  0.083 

H 
bd  =  4.554- 

385 

V 

H 

45 

0.785 

i  =  0.125 

bo  =  3.212  — 

V 

386 

60 

1.047 

i  =  0.167 

bd  =  2.S47- 

387 

V 

75 

1.309 

•£%  =  0.208 

H 
bd  =  2.151  — 

388 

V 

H 

90 

I-57I 

i  =  0.250 

bS  =  1.887- 

V 

389 

H 

105 

T.833 

-ff  =  0.292 

V 

390 

120 

2.094 

1  =  0.333 

ff 

b8  =  1.568- 

V 

391 

H 

135 

2-356 

1  =  0.375 

bd  =  1.463- 

392 

ff 

150 

2.618 

T2   =  0.417 

**=  1-375- 

393 

H 

165 

2.880 

tt  =  0.458 

b8  =1.320- 

394 

180 

3  142 

1  =  o  500 

bS  =  1.260^ 

^ 

395 

195 

3.403 

H  =  °-54i 

,,                H 

bS  =  i.  221  — 

V 

396 

210 

3.665 

A  =  0.583 

r  fr    rRR 

397 

V 

ff 

240 

4.189 

*  -  0.667 

bS  =  1.128- 
» 

398 

// 

270 

4.712 

f  =  0.750 

^5  =  1.089- 

399 

V 

ff 

300 

5-236 

f  =  0.833 

68  =  1.045  — 

400 

?y 

I3O  BELTS  AND  PULLEYS. 

Example. — A  leather  belt  \  inch  thick,  running  over 
a  leather-covered  pulley,  transmits  a  force  of  20  horse- 
power at  a  velocity  of  15  feet  per  second.  Required 
the  width  of  the  belt  for  single  and  double  rawhide- 
lacing,  assuming  that  the  belt  embraces  an  arc  of  the 
pulley  equal  to  2.880  circular  measure.  From  formula 
(346)  we  have 

b  X  i  =  2.167  X  ~,          6  =  2.167  X  ^y  X  4, 

or,  for  single  rawhide-lacing, 

*=;  11.557"=  HiV- 
Formula  (378)  gives 

*  X  i  =  1.898  X  ~,          b=  1.898  X  ~  X  4, 

or,  for  double  rawhide-lacing, 

b  —  10.123*  =  io|". 

Example. — A  leather  belt  running  over  leather-cov- 
ered pulleys  is  -£%  inch  thick  and  12  inches  wide. 
Required  the  velocity  at  which  the  belt  can  transmit  a 
force  of  10  horse-power,  assuming  a  •=.  45°,  and  that 
the  belt  has  a  double  leather-lacing.  We  have  from 
formula  (354) 

10  4.928  X  10 

12  X  A  -  4-928  X  -,          fr «     Ux-^-, 

or  v  —  18.77  ft.  per  second. 


LEATHER-COVERED  PULLEYS.  131 

Example.  —  A  leather  belt  (with  a  riveted  joint) 
running  over  leather-covered  pulleys  is  16  inches  wide 
and  T3¥  inch  thick  ;  the  arc  embraced  by  the  belt  on 
the  smaller  pulley  is  150°,  and  the  velocity  of  the  belt 
10  feet  per  second.  It  is  required  to  determine  the 
horse-power  which  can  be  transmitted  by  the  belt. 
From  formula  (393)  we  have 


or  H  —  21.82. 

By  substituting  #  =  •£•%  in  formulas  (251),  (267), 
(283),  (299),  (315),  (331),  (347),  (363),  (379),  and  (395), 
successively,  we  obtain  the  following  formulas  : 

When  a  =  180°  and  <S  —  ^.", 

Single  leather-lacing,      b  =  o.oi867*.  ....  (401) 

Single  rawhide-lacing,    b  =  0.01727*  ....  (402) 

'  Double  leather-lacing,    b  =  O.oi6i7*.  ....  (403) 

Double  rawhide-lacing,  b  =  0.0151/1  ....  (404) 

Riveted  joint,  b  =  0.01057*.  ....  (405) 

TT 

Single  leather-lacing,      b  =  10.208  —  .....     (406) 

TT 

Single  rawhide-lacing,    b  ==    9.481—  .....     (407) 


132  BELTS  AND  PULLEYS. 

TT 

Double  leather-lacing,    b—    8.850--.  .     .     .     .     (408) 


TT 

Double  rawhide-lacing,  b  —    8.297— (409) 


TT 

Riveted  joint,  b  —     5.760— (410) 


By  substituting  S  —  •£%  in  formulas  (248),  (264),  (280), 
(296),  (3 1 2),  (328),  (344),  (360),  (376),  and  (392),  succes- 
sively, the  following  formulas  may  be  obtained  • 

When  a  =  135°  and  8  =  ^y, 

Single  leather-lacing,      b  =  0.02  i$P.  ....  (411) 

Single  rawhide-lacing,    b  =  o.O2OoP.  ....  (412) 

Double  leather-lacing,    b  —  0.0187/1  ....  (413) 

Double  rawhide-lacing,  b  =  0.0175/1  ....  (414) 

Riveted  joint,  b  —  O.OI22P.  ....  (415) 

TT 

Single  leather-lacing,     b=    11.845 — (4^) 

TT 

Single  rawhide-lacing,  b  =    10.990 — (417) 

TT 

Double  leather-lacing,  b  —    10.258— 


LEATHER-COVERED  PULLEYS.  133 

TT 

Double  rawhide-lacing,        b  =  9.632  —  .     .     .     .     (419) 

TT 

Riveted  joint,  b  =  6.688—.     .     .     .     (420) 

Example.  —  A  leather  belt,  running  over  leather-cov- 
ered pulleys,  transmits  a  force  of  600  pounds.  The 
pulleys  are  of  equal  diameters  (a  =  180°)  and  the  thick- 
ness of  the  belt  is  -£%  inch.  Required  the  width  of 
the  belt  for  double  leather-lacing.  We  have  from 
formula  (403) 

b  =  0.0161  X  600, 
b  =  9.66"  -  9ft'. 


Example.  —  A  g^-inch  leather  belt,  running  over  two 
equal  leather-covered  pulleys,  transmits  a  force  of  15 
horse-power  at  a  velocity  of  10  feet  per  second. 
Required  the  width  of  the  belt  for  a  riveted  joint. 

Formula  (410)  gives 

b  =  5.760  x  )|, 

*  -  8.64"  =  8ft". 

Example.  —  A  ^7¥-inch  leather  belt  (double  rawhide- 
lacing),  running  over  leather-covered  pulleys,  transmits 
a  force  of  600  pounds.  The  arc  embraced  by  the  belt 
on  the  smaller  pulley  is  135°.  Required  the  width  of 
the  belt.  From  formula  (414)  we  have 

b  =  0.0175  X  600, 
b  =  10.50"  =  ioi". 


134  BELTS  AND  PULLEYS. 

Example.  —  A  leather  belt  -^  inch  thick,  running  over 
leather-covered  pulleys,  transmits  a  force  of  15  horse- 
power at  a  velocity  of  10  feet  per  second.  It  is  re- 
quired to  determine  the  width  of  the  belt,  for  single 
leather-lacing,  taking  a  =  135°.  Formula  (416)  gives 

b=  11.845  X  ~» 

b  =  17.77"  =  17**". 

Example.  —  A  leather  belt  -£%  inch  thick  and  20  inches 
wide,  running  over  leather-covered  pulleys,  transmits  a 
force  of  20  horse-power.  The  arc  embraced  by  the 
belt  on  the  smaller  pulley  is  135°.  It  is  required  to  de- 
termine the  velocity  at  which  the  belt  can  be  driven  for 
double  rawhide-lacing.  We  have  from  formula  (419) 

20  9.632  X  20 

20  =  9.632    X   --,  —  ---  , 

or  v  =  9.632  =  9T5¥  feet  per  second. 

The  following  tables,  calculated  from  formulas  (401) 
to  (420),  give  the  forces  in  pounds  (P)and  the  values  of 
the  horse-power  divided  by  the  velocity  in  feet  per 


-J  corresponding  to  different  widths  (from  I 

inch  to  30  inches)  of  ^-inch  -leather  belts  running  over 
leather-covered  pulleys  for  a  =  180°  and  a  —  135°  for 
each  of  the  five  methods  of  joint-fastening  given  above  : 


LEA  THER-  CO  VERED  P  ULLE  V  51 


135 


TABLE  OF  WIDTHS   OF  LEATHER    BELTS  OVER  LEATHER-COVERED 
PULLEYS,  WHEN  a  =  180°  AND  d  =  -f^".     From  Formulas  (401)- 

(405). 


Width 

/*,  single 

P.  single 

P,  double 

P,  double 

P,  riveted 

No. 

inches. 

lacing-. 

hieing. 

lacing. 

lacing. 

joint. 

I 

53-88 

58.04 

62.15 

66.27 

95-51 

I 

I* 

80.82 

87.06 

93-23 

99.40 

I43-27 

2 

2 

107.76 

116.08 

124.30 

132.54 

191.02 

3 

2* 

134.70 

145-10 

I55.38 

165.67 

238.78 

4 

3 

161.64 

174.11 

186.45 

198.81 

286.53 

5 

3i 

188.58 

203.  13 

217.53 

23L94 

334-29 

6 

4 

215.52 

232.15 

248.60 

265.08 

382.04 

7 

44 

242  .  46 

261.17 

279.68 

298.21 

429.80 

8 

5 

269  .  40 

290.19 

310.75 

33L35 

477-56 

9 

54 

296.34 

319.21 

341.83 

364.48 

525.31 

10 

6 

323.28 

348.23 

372.90 

397.61 

573-07 

ii 

7 

377.16 

406.27 

435-05 

463.88 

668.58 

12 

8 

.  43L03 

464.31 

497.20 

530.15 

764.09 

13 

9 

484.91 

522.34 

559-35 

596.42 

859.60 

14 

10 

538.79 

580.38 

621.50 

662.69 

955.11 

15 

ii 

592.67 

638.42 

683.65 

728.96 

1050.62 

16 

12 

646.55 

696.46 

745.80 

795-23 

1146.13 

17 

14 

754-31 

812.54 

870.10 

927.77 

1337.15 

18 

16 

862.07 

928.61 

994-40 

1060.31 

1528.18 

19 

18 

969.83 

1044.69 

1118.71 

1192.84 

1719.20 

20 

20 

1077.59 

1160.77 

1243.01 

1325.38 

1910.22 

21 

22 

II85-34 

1276.84 

1367.31 

1457.92 

2101.24 

22 

'24 

1293.10 

1392.92 

1491.61 

1590.46 

2292.26 

23 

26 

1400.86 

1509.00 

1615.91 

1723.00 

2483.29 

24 

28 

1508.62 

1625.07 

1740    21 

1825.53 

2674.31 

25 

30 

1616.38 

I74LI5 

1864.51 

1988.07 

2865.33 

26 

136 


BELTS  AND  PULLEYS. 


TABLE  OF  WIDTHS  OF  LEATHER  BELTS  OVER  LEATHER-COVERED 
PULLEYS,  WHEN  a  —  180°  and  d  =  £$".  From  Formulas  (406)- 
(410). 


Width 
in 
inches. 

—  ,  single 

leather- 
lacing. 

H     . 
—  ,  single 

rawhide- 
lacing. 

TT 

—  ,  double 

V 

leather- 
lacing. 

jj 

—  ,  double 

V 

rawhide- 
lacing. 

H    . 
—  ,  riveted 

V 

joint. 

No. 

I 

0.0980 

0.1055 

O.II30 

0.1205 

0.1736 

I 

I| 

o.  1469 

0.1582 

0.1695 

O.l8o8 

o  .  2604 

2 

2 

0.1959 

0.2109 

0.2260 

0.2410 

0.3472 

3 

a* 

0.2449 

0.2637 

0.2825 

0.3013 

0.4340 

4 

3 

0.2939 

0.3164 

0.3390 

0.3616 

0.5208 

5 

3i 

0.3429 

0.3692 

0-3955 

0.4218 

0.6076 

6 

4 

0.3918 

0.4219 

0.4520 

0.4821 

0.6944 

7 

4i 

0.4408 

0,4746 

0.5085 

0.5-424 

0.7812 

8 

5 

0.4898 

0.5274 

0.5650 

o  .  6026 

0.8681 

9 

5* 

0.5388 

0.5801 

0.6214 

0.6629 

0.9549 

10 

6 

0.5878 

0.6328 

0.6779 

0.7231 

1.0417 

ii 

7 

0.6857 

0.7383 

0.7909 

0.8437 

I.2I53 

12 

8 

0.7837 

0.8438 

0.9039 

o  .  9642 

1.3889 

13 

9 

0.8817 

0.9493 

.0169 

1.0847 

1.5625 

14 

10 

0.9796 

1.0547 

.1299 

1.2052 

I.736I 

15 

ii 

1.0776 

I  .  1602 

.2429 

1.3258 

1.9097 

16 

12 

1-1755 

1.2657 

•3559 

1.4463 

2.0833 

17 

14 

I.37I5 

1.4766 

•5819 

1.6873 

2.4306 

18 

16 

1.5674 

1.6876 

.8078 

1.9284 

2.7778 

19 

18 

1.7633 

1.8985 

2.0338 

2.1694 

3.1250 

20 

20 

1.9592 

2  .  1095 

2.2598 

2.4105 

3.4722 

21 

22 

2.1552 

2.3204 

2.4858 

2.6515 

3.8194 

22 

24 

2-35II 

2-53I4 

2.7118 

2.8926 

4.1667 

23 

26 

2.5470 

2.7423 

2.9377 

3.1336 

4.5139 

24 

28 

2  .  7429 

2.9532 

3-1637 

3-3747 

4-.  86  1  1 

25 

30 

2.9389 

3.1642 

3.3897 

3.6157 

5.2083 

26 

LEA  THER-  CO  VERED  P  ULLE  YS. 


137 


TABLE  OF  WIDTHS  OF  LEATHER  BELTS  OVER  LEATHER-COVERED 
PULLEYS,  WHEN  a  —  135°  AND  d  —  -£%".  From  Formulas  (411)- 
(415). 


Width 
in 
inches. 

P,  single 
leather- 
lacing. 

P,  single 
rawhide- 
lacing. 

P,  double 
leather- 
lacing. 

P,  double 
rawhide- 
lacing. 

P,  riveted 
joint. 

No. 

I 

46.45 

50.05 

53-62 

57.11 

82.24 

I 

I* 

69.67 

75.08 

80.43 

85.67 

123.36 

2 

2 

92.89 

IOO.IO 

107.24 

114.22 

164.47 

3 

2* 

116.12 

125.13 

134.05 

142.78 

205  .  59 

4 

3 

139-34 

150.15 

160.86 

I7L33 

246.71 

5 

3i 

162.56 

175.18 

187.67 

199.89 

287.83 

6 

4 

185.79 

200.20 

214.48 

228.44 

328*95 

7 

4i 

209.01 

225.23 

241.29 

257.OO 

370.07 

8 

5 

232.23 

250.25 

268.10 

285.55 

411.  18 

9 

5* 

255.46 

275.28 

294.91 

314.11 

452.30 

10 

6 

278.68 

3OO.3O 

321.72 

342  .  66 

493.42 

ii 

7 

325.13 

350.35 

375-34 

399-77 

575.66 

12 

8 

371.57 

4OO.4O 

428.95 

456.88 

657.89 

13 

9 

418.02 

450.45 

482.57 

513.99 

740.13 

14 

10 

464.47 

500.50 

536.19 

571.10 

822.37 

15 

ii 

510.91 

550-55 

589-81 

628.21 

904  .  60 

16 

12 

557.36 

6OO  .  60 

643.43 

685.32 

986.84 

17 

14 

650.26 

700  .  70 

750.67 

799-54 

1151.32 

18 

16 

743.15 

800  .  80 

857.91 

913.76 

1315.79 

19 

18 

836.04 

900.90 

965.15 

1027.98 

1480.26 

20 

20 

928.94 

IOOI.OO 

1072.39 

1142.20 

1644.74 

21 

22 

•  1021.83 

IIOI.IO 

1179.62 

1256.42 

1809.21 

22 

24 

1114.72 

1201.20 

1286.86 

1370.64 

1973.68 

23 

26 

1207.62 

I3OI.3O 

1394.  10 

1484.87 

2138.16 

24 

28 

1300.51 

1401  .  40 

1501.34 

1599.09 

2302.63 

25 

30 

1393.40 

1501.50 

1608.58 

I7I3-3I 

2467.10 

26 

138 


BELTS  AND  PULLEYS. 


TABLE  OF  WIDTHS  OF  LEATHER  BELTS  OVER  LEATHER  COVERED 
PULLEYS,  WHEN  a  —  135°  AND  d  =  -gy.  From  Formulas  (416)- 
(420). 


Width 
in 
inches. 

—  ,  single 

leather- 
lacing. 

H    . 
—  ,  single 

V 

rawhide- 
lacing. 

—  ,  double 

V 

leather- 
lacing. 

—  ,  double 

V 

rawhide- 
lacing. 

—  ,  riveted 
•v 
joint. 

No. 

I 

0.0844 

0.0910 

0.0975 

0.1038 

0.1495 

I 

ri 

O.I266 

0.1365 

0.1462 

0.1557 

0.2243 

2 

2 

0.1689 

0.1820 

0.1950 

0.2076 

o  .  2990 

3 

2i 

0.2III 

0.2275 

0.2437 

0.2596 

0.3738 

4 

3 

0.2533 

0.2730 

0.2924 

0.3H5 

o  .  4486 

5 

3i 

0.2955 

0.3185 

0.3412 

0.3634 

0.5233 

6 

4 

0.3377 

0.3640 

0.3899 

0.4153 

0.5981 

7 

4* 

0.3799 

0.4095 

0.4387 

0.4672 

0.6728 

8 

5 

O.422I 

0.4550 

0.4874 

0.5191 

0.7476 

9 

Si 

0.4643 

0.5005 

0.5362 

0.5710 

0.8224 

10 

6 

0.5066 

o  .  5460 

0.5849 

0.6229 

0.8971 

ii 

7 

0.5910 

0.6370 

0.6824 

0.7267 

I  .  0466 

12 

8 

0.6754 

0.7280 

0.7799 

0.8306 

1.1962 

13 

9 

0.7598 

0.8189 

0.8773 

0.9344 

1-3457 

14 

1C 

0.8443 

o  .  9099 

0.9748 

.0382 

1.4952 

15 

ii 

0.9287 

1.0009 

1.0723 

.1420 

1.6447 

16 

12 

I.OI3I 

1.0919 

I  .  1698 

.2458 

1.7942 

17 

14 

I.I820 

1.2739 

1.3647 

•4535 

2.0933 

18 

16 

1.3508 

1.4559 

1-5597 

.6611 

2.3923 

19 

18 

I-5IQ7 

1.6380 

1-7547 

1.8688 

2.6914 

20 

20 

1.6885 

1.8199 

1.9496 

2.0764 

2.9904 

21 

22 

1.8574 

2.0019 

2.1446 

2  .  2840 

3.2894 

22 

24 

2.0262 

2.1839 

2.3396 

2.4917 

3.5885 

23 

26 

2.I95I 

2.3658 

2-5345 

2.6993 

3-8875 

24 

28 

2  .  3640 

2.5478 

2.7295 

2  .  9070 

4.1866 

25 

30 

2.5328 

2.7298 

2.9245 

3.U46 

4.4856 

26 

Example. — Required  the  force  in  pounds  which  can 
be  transmitted  by  a  ^inch  leather  belt,  20  inches  wide, 
running  over  two  equal  leather-covered  pulleys,  the 
belt-joint  being  riveted.  From  the  table  on  page  135, 
column  for  riveted  joint,  line  21,  we  have 


P=  1910.22  pounds. 


LEATHER-COVERED  PULLEYS.  139 

Example. — A  ^--inch  leather  belt  running  over  leather- 
covered  pulleys,  and  embracing  an  arc  of  135°  on  the 
smaller  pulley,  transmits  a  force  of  1000  pounds.  It  is 
required  to  determine  the  proper  width  for  the  belt  for 
single  rawhide-lacing.  The  table  on  page  137,  column 
for  single  rawhide-lacing,  line  21,  gives,  corresponding 
to  P  =  1001.10  pounds,  a  width  of 

b  =  20". 

Example. — A  ^-inch  leather  belt,  running  over  two 
equal  leather-covered  pulleys  at  a  velocity  of  10  feet 
per  second,  transmits  a  force  of  22  horse-power.  Re- 
quired the  width  for  the  belt  for  a  double  leather-lac- 
ing. We  have  in  this  case  —  =  —  =  2.2,  and  the 
table  on  page  136,  column  for  double  leather-lacing,  line 

TT 

21,  gives  for —  =  2.2598  a  belt-width  of 

6  =  20" 

Example. — A  leather  belt  -^  inch  thick  and  28  inches 
wide,  running  over  leather-covered  pulleys  and  embrac- 
ing an  arc  of  135°  on  the  smaller  pulley,  transmits  a 
force  of  25  horse-power.  It  is  required  to  determine 
the  velocity  at  which  the  belt  can  be  driven  for  a 
double  rawhide-lacing.  From  the  table  on  page  138, 
column  for  double  rawhide-lacing,  line  25,  we  have 

H      25  25 

—  2.9070,        v  = 


v   '  -  2.9070' 


I4O  BELTS  AND  PULLEYS. 

or  v  —  8.60  feet  per  second. 

Example.  —  A  leather  belt  -^  inch  thick  and  28  inches 
wide,  running  over  leather-covered  pulleys  and  embrac- 
ing an  arc  of  135°  on  the  smaller  pulley,  is  driven  at  a 
velocity  of  8.60  feet  per  second.  It  is  required  to  de- 
termine the  horse-power  which  can  be  transmitted  by 
the  belt,  the  joint-fastening  being  a  double  rawhide-lac- 
ing. From  the  table  on  page  138,  column  for  double 
rawhide-lacing,  line  25,  we  have 

TT  TT 

=  2'9°7°>        H  =  8.60  X  2.9070, 


v 


or 


§  12.     Vulcanized-rubber  Belts. 

Vulcanized-rubber  belts  are  usually  made,  as  ex- 
plained in  §  8,  by  placing  one  or  more  layers  of  cotton 
duck  between  layers  of  vulcanized  rubber.  The  num- 
ber of  these  layers  is  indicated  by  the  term  ply  :  thus 
a  one-ply  belt  contains  one  layer  of  duck,  a  three-ply 
belt  contains  three  layers,  etc.  The  thickness  of  each 
layer  of  duck  varies  more  or  less  according  to  the 
amount  of  material  and  the  force  with  which  the  lay- 
ers are  pressed  together  in  the  manufacture.  We  may, 
however,  with  sufficient  correctness  for  ordinary  pur- 
poses, take  for  the  average  thickness  of  a  ply  TV  inch. 
A  three-ply  belt  is  therefore  approximately  J  inch  thick, 
a  four-ply  belt  -J-  inch  thick,  etc. 


VULCANIZED-RUBBER  BELTS.  141 

The  strength  of  vulcanized-rubber  belting  seems  to 
be  about  that  of  leather  of  the  same  thickness.  A 
series  of  tests  made  for  the  author  by  Messrs.  Fair- 
banks &  Co.,  on  their  standard  testing-machine,  gave 
for  superior  new  vulcanized-rubber  belting  an  average 
strength  of  nearly  4000  pounds  per  square  inch  of  sec^ 
tion.  A  great  number  of  other  tests  made  by  the 
author  on  ordinary  vulcanized-rubber  belts  which  had 
been  in  practical  use  for  a  short  time  gave  results  es- 
sentially the  same  as  for  leather. 

We  shall  therefore  use  for  the  safe-working  stress  in 
pounds  per  square  inch  for  vulcanized-rubber  belting 
the  following  values,  given  in  §  10: 

Single  leather-lacing,  f=  325  ; 
Single  rawhide-lacing,  /  =  350  ; 
Double  leather-lacing,  /  =  375  ; 
Double  rawhide-lacing,  f  =  400  ; 
Riveted  joint,  f=  575. 


The  coefficient  of  friction  for  vulcanized  rubber  over 
cast-iron  seems  to  be  slightly  greater  than  for  leather 
over  leather-covered  pulleys.*  Since,  however,  rubber 
belts  are  very  seriously  injured  by  slipping  about  their 
pulleys,  and  for  this  reason  greater  care  should  be  taken 
to  prevent  slipping,  we  propose  to  neglect  the  ap- 
parently small  difference  and  take  the  coefficient  equal 

*  See  Appendix  I. 


142 


BELTS  AND   PULLEYS. 


to  that  for  leather  over  leather-covered  pulleys.     We 
have  then 

<p  =  0.45. 

The  formulas  for  widths  of  vulcanized-rubber  belts 
over  cast-iron  pulleys  may  be  copied  directly  from 
those  for  leather  belts  over  leather-covered  pulleys 
without  the  trouble  of  copying  the  preliminary  tables 
and  formulas. 


TABLE  OF  FORMULAS  FOR  VULCANIZED-RUBBER   BELTS   OVER  CAST- 
IRON  PULLEYS. 

Single  Leather- Lacing. 


a.  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A  =  0.083 

bd  =  O.OI464/* 

421 

45 

0.785 

i  =  0.125 

bd  —  O.OIO34/* 

422 

60 

1.047 

i  =  0.167 

££  =  o.oo8i8/> 

423 

75 

1.309 

•ff  =  0.208 

1)8  —  0.00692^ 

424 

90 

I-57I 

i  =  0.250 

£5   nr  0.00606P 

425 

105 

1.833 

A  =  0.292 

M  =  o-oossi/* 

426 

120 

2.094 

i  =  0-333 

^5  =  0.00503^ 

427 

135 

2.356 

1  =  0.375 

b$  •=.  0.00471^ 

428 

150 

2.618 

A  =  0.417 

b$  =  0.00443/5 

429 

I65 

2.880 

«  =  0.458 

<^5  —  O.OO424/3 

430 

180 

3.142 

%  =  o  500 

b8  =  o.  00406  P 

431 

195 

3.403 

H  =  0.541 

b8  =  0.00394^ 

432 

210 

3.665 

A  =  0.583 

^5  =  0.00382^ 

433 

24O 

4.189 

f  =  0.667 

<^5  =  0.00363^ 

434 

270 

4.712 

f  =  0.750 

k$  =  0.0035I/3 

435 

300 

5-236 

I  =  0.833 

bd  =  o.  00335^ 

436 

VULCANIZED-RUBBER  BELTS. 


143 


TABLE  OF   FORMULAS   FOR  VULCANIZED-RUBBER   BELTS  OVER  CAST- 
IRON  PULLEYS. 
Single  Rawhide  Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A  =  0.083 

bd 

=  o.oi36oP 

437 

45 

0.785 

i  =  0.125 

bd 

=  0.00960^ 

438 

60 

1.047 

j  =  0.167 

bd 

=  o.oofj6oP 

439 

75 

1.309 

•fa  —  0.208 

bd 

=  0.00643/3 

440 

90 

I-57I 

i  =  0.250 

bd 

=  o.  00563^ 

441 

105 

1-833 

fa  =  0.292 

ts 

=  0.005  i  i/3 

442 

120 

2.094 

$  =  0.333 

6$ 

=  0.00469/3 

443 

135 

2.356 

1  =  0.375 

bd 

=  0.00437^ 

444 

150 

2.618 

T5¥  =  0.417 

bS 

=  o.oo4iiP 

445 

I65 

2.880 

ii  =  0.458 

bd 

=  0.00394^ 

446 

1  80 

3.142 

{  =  0.500 

bd 

=  0.00377/3 

447 

195 

3-403 

it  =  0.541 

bd 

=  0.00366^ 

448 

210 

3-665 

A  -0.583 

bd 

rz:  O.OO354/5 

449 

240 

4.189 

1-  =  0.667 

bd 

=  0.00337/3 

450 

270 

4.712 

f  =  0.750 

bd 

=:  0.00326^ 

451 

300 

5-236 

1  =  0.833 

bd 

=  o.oo3iijP 

452 

TABLE  OF  FORMULAS  FOR  VULCANIZED-RUBBER   BELTS   OVER   CAST- 
IRON  PULLEYS. 
Double  Leather- Lacing. 


a.  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

.    30 

0.524 

TV  =  0.083 

b$  =  O.OI269/5 

453 

45 

0.785 

i  —  0.125 

bd  =  0.00896^ 

454 

60 

1.047 

1  =  0.167 

b§  =  o.oojogP 

455 

75 

1.309 

fa  =  O.2O8 

bS  =  o.oo6ooP 

456 

90 

I-57I 

i  =  0.250 

bd  =  0.00525^ 

457 

105 

1.833 

^  =  0.292 

bd  =  0.00477^ 

458 

120 

2.094 

i  =  0.333 

bd  =  0.00437/3 

459 

135 

2.356 

l  =  o.375 

bd  =  o.  00408  P 

460 

150 

2  618 

A  =  0-417 

bd  =  o.  00384^ 

461 

I65 

2.880 

H  =  0-458 

bd  =  o.ootfSP 

462 

1  80 

3.142 

i  =  0.500 

bd-=  O.OO352/5 

463 

195 

3.403 

H  =  0-541 

bd  =  O.OO34I/5 

464 

2IO 

3-665 

A  =  0.583 

3^  =  0.0033  \P 

465 

240 

4.189 

1  =  0.667 

bd  =  0.003I5/5 

466 

27O 

4.712 

f  =  0.750 

bd  =  o.  00304  P 

467 

3OO 

5-236 

1  =  0.833 

bd   =  O.OO2QIP 

468 

144 


BELTS  AND  PULLEYS. 


TABLE  OF   FORMULAS   FOR  VULCANIZED-RUBBER  BELTS  OVER  CAST- 
IRON  PULLEYS. 
Double  Rawhide-Lacing. 


a.  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

T2  =  0.083 

b8  —  o.ongoP 

469 

45 

0.785 

i  =  0.125 

bd  =  0.00840^ 

470 

60 

1.047 

J  =  0.167 

bd  =  o.  00665  /* 

471 

75 

1.309 

•^  =  0.208 

68  —  0.00563^ 

472 

90 

I-57I 

i  =  0.250 

65  =  0.00493/3 

473 

105 

1.833 

-JT  —  0.292 

bd  =  0.00448^ 

474 

120 

2.094 

'*  =  0-333 

M  =  o.oo^ioP 

475 

135 

2.356 

f  =  0-375 

6d  =  0.00383^ 

476 

150 

2.618 

TV  =  0-417 

£5  =  o.oo36o/> 

477 

I65 

2.880 

4i  -  0.458 

£5  =  0.00345/7 

478 

180 

3.142 

\  =  0.500 

<55  =  0.00330^ 

479 

195 

3.403 

if  =  0.541 

^<5  —  0.00320^* 

480 

210 

3.665 

&  =  0-583 

<^5  =  O.OO3IO/1 

481 

240 

4.189 

|  =  0.667 

bd  =  0.00295^ 

482 

270 

4.712 

f  =  0.750 

^5  =  0.00285/* 

483 

300 

5.236 

1=0.833 

^5  =  o.  00273^ 

484 

TABLE  OF  FORMULAS  FOR  VULCANIZED-RUBBER   BELTS   OVER   CAST- 
IRON  PULLEYS. 
Riveted  Joints. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

A  =  0.083 

£&  =  0.00828P 

485 

45 

0.785 

i  =  0.125 

bd  =  o.  00584^ 

486 

60 

1.047 

i  =  0.167 

b8  —  o.  00463^ 

487 

75 

1.309 

•ff  =  0.208 

b8  =  0.0039  iP 

488 

90 

I.57I 

i  =  0.250 

bft  —  0.00343/* 

489 

105 

1.833 

^  =  0.291 

bd  =  o.oo^nP 

490 

120 

2.094 

i  =  0.333 

b§  =  O.OO285/* 

491 

135 

2.356 

f  =  0.375 

b8  =  O.Q0266P 

492 

150 

2.618 

TK¥  =  0.417 

bd  =  0.00250/5 

493 

165 

2.880 

tt  =  0.458 

bd  —  o.oo24oP 

494 

180 

3.142 

-J  =  0.500 

bd  —  0.00229/5 

495 

195 

3-403 

H  =  0-541 

bd  =  O.OO222/5 

496 

210 

3.665 

£  =  0.583 

bd  =  0.002I6/* 

497 

240 

4.189 

I  =  0.667 

bd  —  o.  00205  P 

498 

270 

4.712 

f  =  0.750 

bd  —  o.ooigS/* 

499 

3OO 

5.236 

f  =  0.833 

^(5  —  o.ooigo/' 

500 

VULCANIZED-RUBBER  BELTS. 


145 


TABLE  OF  FORMULAS 


FOR  VULCANIZED-RUBBER  BELTS  OVER 
IRON  PULLEYS. 

Single  Leather-Lacing. 


CAST- 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

TV  =  0 

083 

bo 

=  8 

H 
.052  — 

V 

501 

45 

0.785 

i  =  o 

125 

bo 

=  5 

•687? 

502 

60 

1.047 

i  =  o 

167 

bo 

-  4 

H 
•499- 

503 

75 

1.309 

Jr  =  o 

208 

bd 

—  3 

.806^ 

V 

504 

90 

I.57I 

i  =  o 

250 

bd 

=  3 

H 
•333- 

505 

105 

1.833 

*  =  o 

292 

bd 

—  3 

H 

•031- 

506 

120 

2.094 

l-o 

333 

bd 

=  2 

*? 

507 

135 

2.356 

l-o 

375 

bd 

=  2 

H 

.591- 

508 

150 

2.618 

*  =  o 

4*1 

bd 

=  2 

H 

•437- 

z/  7" 

509 

165 

2.880 

tt  =  o 

458 

bd 

=  2 

•332- 
z> 

510 

180 

3.142 

|-o 

500 

bd 

=  2 

.233^ 

» 

511 

195 

3-403 

H=o 

54i 

bd 

=  2 

.^2 

z^ 

512 

210 

3.665 

A  =  o 

583 

bd 

=  2 

H 
.101  — 
z> 

513 

240 

4.189 

»  =  0 

667 

bd 

—   I 

H 

•997- 

514 

270 

4.712 

f  =  Q 

750 

bd 

=   I 

H 
•93i- 

515 

300 

5-236 

f  =  o 

833 

bd 

—  j 

•8«f 

516 

10 


146 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  VULCANIZED-RUBBER  BELTS  OVER   CAST- 
IRON  PULLEYS. 

Single  Rawhide- Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

jV  =  0.083 

bd 

H 

V 

517 

45 

0.785 

i  =  0.125 

bd 

^^  5  *  280  — 

518 

60 

1.047 

£  =  0.167 

bd 

TT 
=    4.I8O  — 

519 

75 

1.309 

-^  =  0.208 

bd 

H 

=  3-537- 

520 

90 

I.57I 

£  =  0.250 

bd 

II 

=  3.097- 

521 

105 

1.833 

•ff  =  0.292 

bd 

=  2.8ll| 

522 

120 

2.094 

*  =  0.333 

bd 

=  2.580^ 

523 

135 

2.356 

1=0.375 

bd 

=  2.404- 

524 

150 

2.618 

*  =  0.417 

bd 

=  2.261  — 

525 

I65 

2.880 

H  =  o  458 

bd 

—  2.167  — 

526 

1  80 

3.142 

J^o^o 

bd 

V 

527 

195 

3-403 

if  =  0.541 

bd 

—  2.013  — 

528 

210 

3.665 

•&  =  0.583 

bd 

H 

=  1.947- 

529 

240 

4.189 

f  =  0.667 

bd 

=  i.854f 

530 

270 

4.712 

f  =  0.750 

bd 

H 

-1.793- 

531 

300 

5.236 

f  =  0.833 

bd 

=  1.701- 

532 

VULCANIZED-RUBBER  BELTS. 


147 


TABLE  OF  FORMULAS  FOR  VULCANIZED-RUBBER  BELTS  OVER   CAST- 
IRON  PULLEYS. 

Double  Leather-Lacing. 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

H 

30 

0.524 

•&  =  0.083 

bd  =  6.980- 

V 

533 

H 

45 

0.785 

i  =  0.125 

bd  —  4.928- 

534 

ff 

60 

1.047 

£  -  0.167 

bd  =  3.900  — 

535 

75 

1.309 

•ft  =  0.208 

#5  =  3.300— 

536 

90 

I.57I 

i  =  0.250 

<^5  =  2.888- 

537 

105 

1.833 

•fa  —  0.292 

bd  —  2.624  — 

7/ 

538 

// 

120 

2.094 

t  =  0.333 

£o  =  2.404  — 

539 

77 

H 

135 

2.356 

1  -  0.375 

bd  =  2.244- 

540 

jf 

150 

2.6*9 

^  =  0.417 

541 

ff 

,   I65 

2.880 

tt=  0.458 

bd  =  2.024  — 

542 

ff 

180 

3.142 

-J-  ±=  0.500 

bd  =  1.936  — 

543 

195 

3.403 

M  =  0.541 

ff 
bd  =  1.876- 
» 

544 

// 

2IO 

3.665 

•h  =  0.583 

^  =  I.82I- 

545 

Z/ 

ff 

240 

4.189 

f  ±=  0.667 

bd  —  1.733- 

546 

^ 

77 

270 

4.712 

f  -  0.750 

M  =  1.672- 

547 

ff 

30O 

5.236 

f  =  0.833 

^5  =  i  .601  — 

548 

148 


BELTS  AND  PULLEYS. 


TABLE  OF  FORMULAS  FOR  VULCANIZED-RUBBER  BELTS  OVER 
IRON  PULLEYS. 

Double  Rawhide- Lacing. 


CAST- 


a  in 
degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

30 

0.524 

-1%  =  0.083 

bd  = 

6.545f 

549 

45 

0.785 

•£•  =  0.125 

bd  = 

A     ^ 
4.620— 

V 

550 

60 

1.047 

i  =  0.167 

bd  = 

3.658^ 

V 

551 

75 

1.309 

-f^  =.  O.208 

bd  = 

V 

552 

90 

I.57I 

i  =  0.250 

bd  = 

2.712^ 

553 

105 

1.833 

-fa  =  0.292 

bd  = 

2.464  — 

554 

1  20 

2.094 

i  =  0.333 

bd  -^ 

H 

2.225- 

555 

135 

2.356 

1  =  0.375 

bd  = 

H 
2.107  — 

556 

150 

2.618 

A  =  0.417 

*** 

1.980— 

557 

165 

2.880 

tt=  0.458 

M  = 

TT 

1.898- 

558 

1  80 

3.142 

i  =  0.500 

M  = 

mif 

559 

195 

3.403 

ft  =  0.541 

^  = 

t-j,6.f 

560 

210 

3-665 

T2    =   0.583 

/^6V  = 

I  •  705  - 

56i 

240 

4.189 

|   =  0.667 

bd  = 

1.623— 

562 

270 

4.712 

f  =  0.750 

bd  = 

1  .  568  — 
S5> 

563 

300 

5.236 

f  —  0.833 

bd  = 

H 
1.502- 

564 

VULCANIZED-RUBBER  BELTS. 


149 


TABLE  OF   FORMULAS 


FOR  VULCANIZED-RUBBER  BELTS  OVER 
IRON  PULLEYS. 

Riveted  Joints. 


CAST- 


a  in 
degrees. 

o  in  circular 
measure. 

a  in  fractions  of 
circumference. 

Formula. 

No. 

H 

30 

0.524 

A-  —  0.083 

bd 

=  4-554- 

565 

ff 

45 

0.785 

I  -  0.125 

bd 

=  3.212  — 

566 

H 

60 

1.047 

J  =  0.167 

bd 

=  2.547  — 

OH"  v 

567 

H 

75 

1.309 

^  =  0.208 

bd 

568 

IT 

90 

I-57I 

i  =  0.250 

bd 

=  1.887- 

569 

V 

H 

105 

1.833 

•fa  =  0.292 

bd 

=  1.711- 

570 

V 

-  i     68^ 

120 

2.094 

i  =  0.333 

bd 

571 

135 

2.356 

1  =  0.375 

bd 

=  i.463f 

572 

ff 

150 

2.618 

y5^  —  °'4I7 

bd 

—  1-375- 

573 

V 

If 

.    I65 

2.880 

it  -0.458 

bd 

—  1.320  — 

574 

V 

ff 

180 

3.142 

i  =  0.500 

bd 

=  1.260  — 

V 

575 

H 

195 

3.403 

M  =  0.541 

bd 

=  I.22I— 
V 

576 

210 

3.665 

A  =  0.583 

bd 

=  1.188^ 

577 

V 

IT 

240 

4  189 

i-  =  0.667 

bd 

=  1.  128- 

578 

z/ 

H 

270 

4.712 

t  =  0.750 

bd 

=  1.089- 

579 

ff 

300 

5-236 

1  =  0.833 

bd 

580 

150  BELTS  AND  PULLEYS. 

The  formulas  for  vulcanized-rubber  belts  -^  inch 
thick  (say  three-ply)  over  cast-iron  pulleys  are  as 
follows  : 

When  a=  180°, 

Single  leather-lacing,      b  —  O.OI86P;  ....  (581) 

Single  rawhide-lacing,    b  =  O.OJ72/;  ....  (582) 

Double  leather-lacing,    b  —  o.oi6iP;  ....  (583) 

Double  rawhide-lacing,  £  =  o.oi5iP;  ....  (584) 

Riveted  joint,                   b  =  0.0105^.  ....  (585) 

TT 

Single  leather-lacing,       £=10.208 — ;       .     .     .     (586) 

v 

TT 

Single  rawhide-lacing,     £  =    9.481-—;      .     .     .     (587) 

Double  leather-lacing,     b—    8.850 — ;       .     .     .     (588) 

v 

Double  rawhide-lacing,    b  =    8.297-— ;       .     .     .     (589) 
Riveted  joint,  b  =    5.760—.        .     .     .     (590) 

When  a  —  135°, 

Single  leather-lacing,      £  =  0.0215^;  ....  (591) 

Single  rawhide-lacing,     £  =  O.O2OO/;  ....  (592) 

Double  leather-lacing,    £  =  o.oi87/>;  ....  (593) 

Double  rawhide-lacing,  £  =  0.0175^;  ....  (594) 

Riveted  joint,                   £  =  0.0122/1  ....  (595) 

TT 

Single  leather-lacing,       £=11.845 — ;       .     .     .     (596) 

TT 

Single  rawhide-lacing,     £  =  10.990 — ;       .     .     .     (597) 


VULCANIZED-RUBBER  BELTS. 


TT 

Double  leather-lacing,    b  =  10.258 — ; 

TT 

Double  rawhide-lacing,   b  =    9.632- — ; 


Riveted  joint, 


6=    6.668 


H 

v  ' 


(598) 

(599) 
(600) 


TABLE  OF  WIDTHS  OF  VULCANIZED-RUBBER  BELTS  OVER  CAST-IRON 
PULLEYS,  WHEN  a.  =  180°  AND  8  =  •fe".     From  Formulas  (5 81)- 

(585). 


Width 
in 
inches. 

/*,  single 
leather- 
lacing. 

P,  single 
rawhide- 
lacing. 

P,  double 
leather- 
lacing. 

P,  double 
rawhide- 
lacing. 

P,  riveted 
joints. 

No. 

I 

53.88 

58.04 

62.15 

66.27 

95-51 

I 

I* 

80.82 

87.06 

93.23 

99-40 

143.27 

2 

2 

107.76 

116.08 

124.30 

132.54 

191.02 

3 

*t 

I34-70 

145.10 

155.38 

165.67 

238.78 

4 

3 

161.64 

174.11 

186.45 

198.81 

286.53 

5 

31 

188.58 

203.  13 

217.53 

231.94 

334-29 

6 

4 

215-52 

232.15 

248  .  60 

265.08 

382.04 

7 

4i 

242  .  46 

261.17 

279.68 

298.21 

429.80 

8 

5 

269.40 

290.  19 

310.75 

331-35 

477.56 

9 

5i 

296.34 

319-21 

341.83 

364.48 

525.31 

10 

6 

323.28 

348.23 

372.90 

397.61 

573-07 

ii 

7 

377-16 

406.27 

435.05 

463.88 

668.58 

12 

8 

431-03 

464.31 

497.20 

530.15 

764.09 

13 

9 

484.91 

522.34 

559-35 

596-42 

859.60 

14 

10 

538-79 

580.38 

621.  50 

662.69 

955-11 

15 

ii 

592.67 

638  42 

683.65 

728.96 

1050.62 

16 

12 

646.55 

696  .  46 

745.80 

795-23 

1146.13 

17 

14 

754-31 

812.54 

870.  10 

927.77 

1337.15 

18 

16 

862.07 

928.61 

994.40 

1060.31 

1528.18 

*9 

18 

969.83 

1044.69 

1118.71 

1192.84 

1719.20 

20 

20 

1077-59 

1160.77 

1243.01 

I325-38 

1910.22 

21 

22 

1185-34 

1276.84 

1367.31 

1457.92 

2101.24 

22 

24 

1293.  10 

1392-92 

1491.61 

I590-46 

2292.26 

23 

26 

1400.86 

1509.00 

1615.91 

1723.00 

2483.29 

24 

28 

1508.62 

1625.07 

1740.21 

1825.53 

2674.31 

25 

30 

1616.38 

I74I-I5 

1864.51 

1988.07 

2865.33 

26 

152 


BELTS  AND  PULLEYS, 


TABLE  OF  WIDTHS  OF  VULCANIZED-RUBBER  BELTS  OVER  CAST-IRON 
PULLEYS,  WHEN  a  =  180°  AND  6  =  -£%".  From  Formulas  (586)- 
(590). 


Width 
in 
inches. 

H 
—  ,  single 
leather- 
lacing. 

H 
—  ,  single 

rawhide- 
lacing. 

H 

~,  double 

leather- 
lacing. 

H 
—  ,  double 

rawhide- 
acing. 

H 
—  ,  riveted 

joint. 

No. 

I 

0.0980 

0.1055 

O.II30 

0.1205 

0.1736 

I 

I* 

0.1469 

0.1582 

0.1695 

0.1808 

0.2604 

2 

2 

0.1959 

0.2109 

0.2260 

0.2410 

0.3472 

3 

*i 

0.2449 

0.2637 

0.2825 

0.3013 

0.4340 

4 

3 

0.2939 

0.3164 

0.3390 

0.3616 

0.5208 

5 

31 

0.3429 

0.3692 

0-3955 

0.4218 

0.6076 

6 

4 

0.3918 

0.4219 

0.4520 

0.4821 

0.6944 

7 

4* 

0  .  4408 

0.4746 

o  5085 

0.5424 

0./8I2 

8 

5 

0.4898 

0.5275 

0.5650 

0.6026 

0.8681 

9 

54 

0.5388 

0.5801 

0.6214 

0.6629 

0-9549 

10 

6 

0.5878 

0.6328 

0.6779 

0.7231 

1.0417 

ii 

7 

0.6857 

0.7383 

0.7909 

0.8437 

I.2I53 

12 

8 

0.7837 

0.8438 

0.9039 

o  .  9642 

1.3889 

13 

9 

0.8817 

0.9493 

I  .0169 

.0847 

1.5625 

14 

10 

0.9796 

1-0547 

I  .1299 

.2052 

1.7361 

15 

ii 

1.0776 

I.  1602 

1.2429 

.3258 

1.9097 

16 

12 

•1755 

1.2657 

1-3559 

.4463 

2.0833 

17 

14 

.3715 

1.4766 

I.58I9 

.6873 

2  .  4306 

18 

16 

.5674 

1.6876 

1.8078 

.9284 

2.7778 

19 

18 

.7633 

1.8985 

2.0338 

2.1694 

3-I250 

20 

20 

•9592 

2.1095 

2.2^98 

2.4105 

3-4722 

21 

22 

•2.1552 

2.3204 

2.4858 

2.6515 

3.8194 

22 

24 

2.35H 

2.53H 

2.7II8 

2.8926 

4.1667 

23 

26 

2.5470 

2.7423 

2-9377 

3.1336 

4.5139 

24 

28 

2.7429 

2.9532 

3-I637 

3.3747 

4.86II 

25 

30 

2.9389 

3.1642 

3o897 

3.6157 

5-2083 

26 

UN: 


J>  ULCANIZED-R  UBBER  BEL  TS 


153 


TABLE  OF  WIDTHS  OF  VULCANIZED-RUBBER  BELTS  OVER  CAST-IRON 
PULLEYS,  WHEN  a  =  135°  AND  d  =  -£%".  From  Formulas  (591)- 
(595). 


Width 
in 
inches. 

/*,  single 
leather- 
lacing. 

/',  single 
rawhide- 
lacing. 

P,  double 
leather- 
lacing. 

/>,  double 
rawhide- 
lacing. 

/>,  riveted 
joint. 

No. 

I 

46.45 

50.05 

53-62 

57-11 

82   24 

I 

Ii 

69.67 

75.08 

80.43 

85.67 

123.36 

2 

2 

92.89 

100.10 

107.24 

114.22 

164.47 

3 

2| 

116.12 

125.13 

134.05 

142.78 

205-59 

4 

3 

139-34 

150.15 

160.86 

I7I-33 

246.71 

5 

3i 

162.56 

175.18 

187.67 

199.89 

287.83 

6 

4 

I35-79 

200  .  20 

214.48 

228.44 

328.95 

7 

4i 

209.01 

225.23 

241.29 

257.00 

370.07 

8 

5 

232.23 

250.25 

268.10 

285.55 

411.18 

9 

5^ 

25S-46 

275.28 

294.91 

314.11 

452.30 

10 

6 

278.68 

300.30 

321.72 

342.66 

493.42 

ii 

7 

325.13 

350.35 

375-34 

399-77 

575-66 

12 

8 

371-57 

400  .  40 

428.95 

456.88 

657.89 

13 

9 

.    418.02 

450.45 

482.57 

513.99 

740.13 

14 

10 

464.47 

500.50 

536.19 

571.10 

822.37 

15 

ii 

510.91 

550.55 

589.81 

628.21 

904  .  60 

16 

12 

557.36 

600.60 

643.43 

685.32 

986.84 

17 

14 

650.26 

700  .  70 

750.67 

799.54 

1151.32 

18 

16 

743-15 

800.80 

857.91 

913-76 

I3I5.79 

19 

18 

836.04 

900.90 

965-15 

1027.98 

1480.26 

20 

20 

928.94 

1001  .00 

1072.39 

1142.20 

1644.74 

21 

22 

1021.83 

IIOI.  10 

1179.62 

1256.42 

1809.21 

22 

,    24 

1114.72 

1201  .20 

1286.86 

1370.64 

1973.68 

23 

26 

1207.62 

1301.30 

1394.10 

1484.87 

2138.16 

24 

28 

1300  .51 

1401.40 

1501.34 

1599.09 

2302.63 

25 

30 

1391-40 

1501.50 

1608.58 

1713-31 

2467.10 

26 

154 


BELTS  AND  PULLEYS. 


TABLE  OF  WIDTHS  OF  VULCANIZED-RUBBER  BELTS  OVER  CAST-IRON 
PULLEYS,  WHEN  a  =  135°  AND  d  =  -fa".  From  Formulas  (596)- 
(600). 


Width 
in 
inches. 

—  ,  single 

leather- 
lacing. 

—  ,  single 

rawhide- 
lacing. 

—  ,  double 

V 

leather- 
lacing. 

IT 

—  ,  double 

V 

rawhide- 
lacing. 

—  ,  riveted 

V 

joint. 

No. 

I 

0.0844 

0.0910 

0.0975 

o.  1038 

0.1495 

I 

I* 

O.I266 

0.1365 

o.  1462 

0.1557 

0.2243 

2 

2 

0.1689 

O.I82O 

0.1950 

0.2076 

0.2990 

3 

2i 

O.2III 

o  2275 

0.2437 

0.2596 

0.3738 

4 

3 

0.2533 

0.2730 

0.2924 

0.3H5 

0.4486 

5 

3i 

0-2955 

0.3185 

0.3412 

0.3634 

0.5233 

6 

4 

0.3377 

0.3640 

0.3899 

0.4153 

0.5981 

7 

41 

0-3799 

o  4095 

0.4387 

0.4672 

0.6728 

8 

5 

0.4221 

0.4550 

0.4874 

0.5191 

0.7476 

9 

5£ 

0.4643 

0.5005 

0.5362. 

0.5710 

0.8224 

10 

6 

0.5066 

0.5460 

0.5849 

0.6229 

0.8971 

ii 

7 

0.59TO 

0.6370 

0.6824 

0.7267 

I  .  0466 

12 

8 

0.6754 

0.7280 

0.7/99 

0.8306 

I.  1962 

13 

9 

0.7598 

0.8189 

0.8773 

0.9344 

1-3457 

14 

10 

0.8443 

o  .  9099 

0.9748 

1.0382 

1.4952 

15 

ii 

0.9287 

I  .  0009 

1.0723 

I  .  1420 

1.6447 

16 

12 

I.OI3I 

1.0919 

1.1698 

1.2458 

I  .  7942 

17 

14 

I.I820 

1.2739 

1.3647 

1.4535 

2.0933 

18 

16 

1.3508 

1-4559 

1-5597 

1.6611 

2.3923 

19 

18 

I.5I97 

1.6380 

1-7547 

i.  8688 

2.6914 

20 

20 

1.6885 

1.8199 

i  .  9496 

2.0764 

2.9904 

21 

22 

1.8574 

2.0019 

2  .  1446 

2.2840 

3.2894 

22 

24 

2.0262 

2.1839 

2  •  3396 

2.4917 

3.5885 

23 

26 

2.I95I 

2.3658 

2-5345 

2.6993 

3-8875 

24 

28 

2  .  3640 

2.5478 

2.7295 

2  .  9070 

4.1866 

25 

30 

2.5328 

2.7298 

2.9245 

3.II46 

4.4856 

26 

Example. — Required  the  width  for  a  vulcanized-rub- 
ber  belt  f  inch  thick  which  will  transmit  a  force  of  1200 
pounds,  the  fastening  being  a  single  rawhide-lacing  and 
the  arc  embraced  by  the  belt  on  the  smaller  pulley 
being  a  =.  90°. 


VULCANIZED-RUBBER  BELTS.  155 

Formula  (441)  gives 

b  X  -  —  0.00563  X  1 200. 
4 

Hence  b  —  0.00563  X  1200  X  -, 

o 

or  b  =  9". 

Example. — Required  the  width  for  the  above  belt 
with  riveted  joint  instead  of  single  rawhide-lacing. 
We  have  from  formula  (489) 

b  X  -  =  0.00343  X  1 200, 
4 

b  —  0.00343  X  1200  X  -, 

or  b  =  5489"  =  Sir- 

Example.—-  A    vulcanized-rubber    belt    ^-inch    thick 

t embraces  an  arc  equal  to  \  the  circumference  of  its 

smaller  pulley,  and  transmits  a  force  of  20  horse-power 

at   a  velocity   of    10   feet   per  second.     Required   the 

proper  width  for  double  leather-lacing. 

Formula  (539)  gives 

I  20 

*  X  4  =    2'4°4  *  — , 
b  —  2.404  X  2  X  4, 
or  b  =  19.232"  --=  1 944". 


I  ?6  BELTS  AND  PULLEYS. 

Example. — A  three-ply  vulcanized-rubber  belt  run- 
ning over  two  equal  pulleys  transmits  a  force  of  1275 
pounds.  Required  the  proper  width  for  single  raw- 
hide-lacing. The  table  on  page  151,  column  for  single 
rawhide-lacing,  line  22,  gives,  corresponding  to  P  = 
1276.84  pounds, 

b  =  22". 

Example. — Given  the  data  H  =  20,  v  =  20,  a  •=. 
135°,  d  —  ^-inch.  Required  the  proper  width  for  the 
belt,  for  single  rawhide-lacing. 

TT 

The  table  on  page  1 54  gives,  corresponding  to  —  =  I , 

a  belt-width  of  II  inches.     (Column  for  single  rawhide- 
lacing,  line  16.) 

Vulcanized-rubber  belts  are  very  rarely  seen  running, 
over  leather  or  rubber  covered  pulleys.    We  may,  how- 
ever, take  for  the  coefficients  of  friction  of  rubber  on 
leather  and  rubber  on  rubber,  respectively, 

cp  —  0.50* 
and  <p  — 0.55. 

The  general  formula  (50)  for  the  cross-section  of  any 
belt  for  a  given  tension  is 

i 
T 

*=/• 

*  Obviously  this  coefficient  may  be  used  for  leather  belts  over 
rubber-covered  pulleys.  See  Appendix  I. 


VULCANIZED-RUBBER  BELTS. 
This  may  be  put  in  the  form 


157 


-—, 


(601) 


and  the  value  of  x  for  each  special  case  determined 
from  the  tensions  T and  /,  as  in  §§  10  and  II.  The  folj 
lowing  table  gives  values  of  x  for  all  cases  likely  to 
occur  in  practice  : 

TABLE  OF  GREATEST  TENSION  FOR  VULCANIZED  RUBBER  BELTS  OVER 
LEATHER  AND  RUBBER-COVERED  PULLEYS. 


jr,  leather- 

x,  rubber- 

x,  leather- 

x,  rubber- 

a  in 
degrees. 

covered 
pulleys. 

covered 
pulleys. 

degrees. 

covered 
pulleys. 

covered 
pulleys. 

30 

4-35 

3-99 

150 

i-37 

I-3I 

45 

3.08 

2.85 

165 

i-3i 

.26 

60 

2.45 

2.28 

1  80 

1.26 

.21 

75 

2.08 

1-95 

195 

1.22 

.18 

90 

1.85 

1-73 

210 

I.I9 

•15 

105 

1.67 

1-57 

240 

I.I4 

.11 

120 

1-54 

1.46 

270 

I.IO 

I.  08 

-33 

1.44 

1.38 

300 

I.  08 

I.  06 

Example.  —  Required  the  proper  width  for  a  vulcan- 
fzed-rubber  belt  \  inch  thick,  and  transmitting  a  force 
of  800  pounds  over  leather-covered  pulleys,  taking  the 
angle  a  =  120°,  and  the  fastening  a  single  rawhide- 
lacing. 

The  table  gives  for  the  value  of  the  variable  coefficient 


and  the  value  of  the  safe-working  stress  for  single  raw- 
hide-lacing is 

7=350. 


158  BELTS  AND  PULLEYS. 

Hence  formula  (60  1)  becomes 

I        800  X  1.54 
<  4  =  350 

or  6  =  14.08"    = 


Example.  —  A  vulcanized-rubber  belt  J  inch  thick, 
running  over  rubber-covered  pulleys,  transmits  a  force 
of  25  horse-power  at  a  velocity  of  10  feet  per  second. 
Required  the  proper  width  for  double  rawhide-lacing, 
the  arc  embraced  by  the  belt  on  the  smaller  pulley 
being  135°. 

From  the  table,          x  —  1.38, 
and  from  page  141,      f  =  400. 

TT 

We  also  have  P  =  550  -7-. 

Substituting  these  values  in  formula  (601)  gives 

b  x  ^  =  550  x  ^  x  1.38  -*•  400. 

Hence  *  =  5goX  25  X-  1.38  X  4 

10  X  400 

or  b  =  18.98"  =  i8f|". 

Example.  —  A  (J-inch  thick)  vulcanized-rubber  belt 
12  inches  wide  runs  over  leather-covered  pulleys,  and 
embraces  an  angle  of  90°  upon  the  smaller  pulley. 
Required  the  force  in  pounds  which  may  be  safely 
transmitted  by  the  belt  with  a  double  rawhide-lacing. 


/t- 

r* 

RIM,    NAVE,   AND  FIXING-KEYS  FOR  PULLEYS.    159 
The  table  gives  x  =  1.85, 

and  we  have  also  /  =  400. 

Hence  formula  (60  1)  gives 


/  I  -  P  x  *' 
K  4  "        400 


l2  x 


X  4QQ 


1.85 


or 


=  648.65. 


§  13.     /?//w,  /Ifaire,  a/76/  Fix  ing-keys  for  Pulleys.* 

The  rim  of  a  pulley  intended  to  carry  a  flat  belt  is 
generally  slightly  rounded  (Figs.  48  and  49),  in  order 
that  the  belt  may  remain  in  the  centre  of  the  pulley- 
face,,  instead  of  working  to  one  side,  as  is  the  case  with 
flat-faced  pulleys.  The  amount  of  this  rounding  (s) 
may  be  taken  equal  to  ¥V  the  width  of  the  belt. 

For  isolated  pulleys  the  face-width  B  is  taken  some- 
what greater  than  the  width  of  the  belt  (£)  ;  often  we 
.take 


(602) 


When,  however,  several  pulleys  are  placed  side  by 
side  in  order  to  receive  alternately  the  same  belt  the 
face  -width  B  should  be  taken  only  very  slightly 
greater  than  the  belt-width  b. 

The  thickness   k  of    the   edge   of   the  rim,  or  the 

*  From  "  Reuleaux." 


i6o 


BELTS  AND  PULLEYS. 


thickness  at  the  ends  of  the  face-width,  may  be  easily 
calculated  from  the  formula 


k    =    0.08    +  -— . 

IOO 


(603) 


High-speed  pulleys  and  those  subjected  to  consider- 
able shock  and  vibration  are  often  provided  with  late- 
ral flanges  cast  on  the  rims,  as  shown  in  Fig.  49,  or  are 
replaced  by  grooved  pulleys  carrying  belts  with  circu- 
lar cross-section  (Fig.  50). 

Example. — Required  the  rim  dimensions  for  an  iso- 


FIG.  48. 


FIG.  49. 


FIG. 


lated  pulley  which  is  to  carry  a  belt  12  inches  wide. 
From  formula  (602)  we  have  for  the  face-width 


£=      X  12  =  15"; 
4 

and  from  formula  (603),  for  the  thickness  of  the  rim  at 
the  edges, 


RIM,   NAVE,   AND  FIXING-KEYS  FOR  PULLEYS.    l6l 

For  the  amount  of  rounding  of  the  pulley-face,  s  = 

—b  =  —  X  12  =  0.6".  The  thickness  of  the  rim  at 
20  20 

the  centre  is,  therefore, 

2k  +  s  =  2  X  0.23  -j-  0.6  =  i .06". 

If  we  wish  to  provide  the  pulley  with  rim-flanges,  as 
in  Fig.  49,  we  have  for  the  height  of  the  flanges  8/&  = 
8  X  0.23  =  1.84",  and  take  the  thickness  of  the  flanges 
equal  to  k. 

Nave. — The  thickness  (w,  Fig.  56)  of  a  pulley-nave 
is  given  by  the  formula 

/  r> 

*£/  =  0.4  + £  +  --,.     .     .     .     (604) 

in  which  d  represents  the  diameter  of  the  shaft  upon 
which  the  pulley  is  keyed,  and  R  the  radius  of  the 
pulley. 

The  length  of  the  nave  should  not  be  taken  less  than 

L  =  2.$ow (605) 

Often  (in  idle  pulleys,  for  example)  the  length  L  is 
taken  equal  to  the  face-width  B  of  the  pulley. 

Example. — A  pulley  of  36  inches  diameter  is  keyed 
upon  a  shaft  of  4  inches  diameter  ;  required  the  nave 
dimensions.  From  formula  (604)  the  thickness  is 

A       18 
w  =  o.4  +     +      =  1427", 


1 62 


BELTS  AND  PULLEYS. 


and  from  formula  (605)  we  have  for  the  length  of  the 
nave 

L  =  2.$ox  1-427  =  3-5675". 

In  idle  pulleys  the  interior  diameter  of  the  nave,  or 
the  eye  of  the  pulley,  is  taken  slightly  greater  than  the 


FIG.  51. 


diameter  of  the  shaft  upon  which  the  pulley  is  to  run  ; 
often  the  eye  of  an  idle  pulley  is  furnished  with  a  coat- 
ing of  bronze  or  white  metal,  in  order  to  diminish  the 
friction. 

Keys. — There  are  three  kinds  of  keys  which  are  used 
to  fix  pulleys  upon  their  arbors :  the  hollow  key  (Fig. 


FIG.  52. 


FIG.  53. 


FIG.  54. 


54),  used  for  light  pulleys;  the  flat  key  (Fig.  52),  used 
for  pulleys  of  medium  size ;  and  the  countersunk  key 
(Fig.  53),  used  for  very  large  and  heavy  pulleys. 


RIM,    NAVE,    AND  FIXING-KEYS  FOR  PULLEYS.    163 

The  width  s  and  thickness  s1  of  the  fixing-key  are 
given  by  the  expressions 


q>       ' 

d 

Sl  ~              ~  10' 

.  .  (6o7) 

and  the  inclination  varies  from  y^-  to  - 

Example.  —  Required  the  width  and  thickness  of  the 
fixing-key  for  the  pulley  of  the  preceding  example, 
in  which  the  diameter  of  the  shaft  is  d  =  4''.  For- 
mulas (606)  and  (607)  give  for  the  required  width  and 
thickness,  respectively, 


and 


=  0.96", 


^  =  0.16  +  -      =  0.56". 
r  10 


FIG.  55. 

Split  pulleys  (Fig.  55)  are  often  used  for  light  work. 
They  offer  the  advantage  of  being  easily  put  up  and 
taken  down  without  interfering  with  the  shaft-hang- 
ing's. With  oullevs  of  this  kind  fixing -kevs  mav  be 


164 


BELTS  AND  PULLEYS. 


dispensed  with,  the  two  parts  of  the  pulley  being 
pressed  upon  the  shaft  by  means  of  the  nuts  a,  a,  with 
sufficient  force  to  prevent  slipping.  For  this  purpose 
the  eye  of  the  pulley  is  made  slightly  less  than  the 
diameter  of  the  shaft  upon  which  the  pulley  is  to  be 
fastened.  When  the  division  passes  through  a  pair  of 


FIG.  56. 

arms,  as  in  the  figure,  each  half  of  the  split  arm  must 
be  as  strong  as  an  entire  undivided  arm,  and  conse- 
quently of  the  same  dimensions  as  the  entire  arms.* 

Weight  of  Pulleys. — The  weights  of  pulleys  can  evi- 
dently be  calculated  from  one  formula  only  approxi- 
mately, since  the  arms,  nave,  etc.,  vary  considerably  in 

*  A  better  and  stronger  form  of  split  pulley  is  represented  in  Fig. 
56.  In  this  case  all  the  arms  are  entire,  and  the  pulley  presents  a 
better  appearance,  as  well  as  a  simpler  form.  According  to  Unvvin 
(see  "Elements  of  Machine  Design,"  §  168),  the  net  section  of  the 
bolt  at  the  rim  should  be  one  quarter  the  section  of  the  rim  plus  \ 
square  inch,  and  that  of  the  bolt  at  the  nave  one  quarter  the  section 
of  the  nave  plus  £  square  inch. 


RIM,   NAVE,   AND  FIXING-KEYS  FOR  PULLEYS. 


different  pulleys.  We  may.  however,  calculate  the 
weights  of  pulleys  with  sufficient  accuracy  for  ordinary 
purposes  from  the  formula 

7?  /  /A 2  /  7?\  3\ 

G  =  (0.163  ?  +  0.015^  j  +  0.00309^)  )b\  .  (6oS) 

in  which  G  is  the  weight  of  the  pulley  in  pounds,  R 
and  b  respectively  the  radius  of  the  pulley  and  width 
of  the  belt. 

£2 

The  following  table  gives  values  of  yj  for  different 

r> 

values  of  -r- : 

TABLE  OF  WEIGHTS  OF  PULLEYS. 


R 
b 

G 

£3 

R 
b 

G 

b* 

R 

b 

G 

b* 

R 

b 

G 

b* 

1.0 

0.181 

2-5 

0.550 

5-0 

1-579 

8.25 

4.111 

.1 

0.202 

2.6 

0.580 

5-2 

1.691 

8.50 

4.378 

.2 

0.223 

2.7 

0.612 

5-4 

1.807 

8.75 

4.657 

•3 

0.244 

2.8 

0.642 

5.6 

1.929 

9.00 

4-947 

•4 

0.266 

2.9 

0.675 

5.8 

2.057 

9-25 

5.250 

•5 

0.289 

3-o 

0.708 

6.0 

2.  190 

9-50 

5.567 

.6 

O.3I2 

3-2 

0.777 

6.2 

2.329 

9-75 

5.895 

•7 

0-335 

3-4 

0.850 

6.4 

2.473 

IO.OO 

6.237 

.8 

0.360 

3-6 

0.926 

6.6 

2.623 

10.25 

6.592 

!-9 

0.385 

3-8 

.007 

6.8 

2.780 

10.50 

6  961 

2.0 

0.4II 

4.0 

.oSq 

7-o 

2-943 

11.00 

7.742 

2.1 

0-437 

4.2 

.180 

7-25 

3-155 

11.50 

8.581 

2.2 

0.464 

4  4 

•273 

7-50 

3-378 

I2.OO 

9.482 

2-3 

0.492 

4.6 

•370 

7-75 

3.6II 

12.50 

10.446 

2.4 

0.520 

4-8 

.472 

8.00 

3.856 

13.00 

n.475 

Example. — The  radius  of  a  pulley  is  16  inches,  and 
the  width  of  the  belt  which  runs  upon  the  pulley  4 
inches  ;  required  the  approximate  weight  of  the  pulley. 

/?       T  6 
Here   ,  =  —  —  4.     From  formula  (608), 


l66  BELTS  AtVD    PULLEYS. 

G  =  (0.163  X  4  +  0.015  X  1 6  +  0.00309  X  64)64, 
G  =  (0.652  -f-  0.240  4~  0.19776)64  =  1.08976  X  64 ; 
or,  G  =  69.74  pounds. 

Example. — Required    the   approximate  weight  of  a 
pulley  for  the  data  R  =  36",  b  —  4^".     In  this  case 

T-  =  ~£  —  8,  and  tf  —  91.125.     From  the  table  we  find 
R  _ 

%  =  3.856. 
Hence      G  =  3.856  X  91.125  =  351.378  pounds. 

§  14.     Arms  of  Pulleys* 

Ordinarily  the  arms  of  pulleys  have  oval  cross-sec- 
tions, the  diameter  in  the  plane  of  the  pulley  being 
twice  the  smaller  diameter.     The  profile  of  such  a  cross- 
K  section  may  be  drawn  by  circle- 

arcs  as  shown  in  Fig.  57.  The 
dotted  circle  is  drawn  on  the 
greater  diameter  h^  of  the  pul- 
\  C1  ley-arm>  and  the  arcs  ab  and 
a'b'  have  their  centres  respec- 

J\    ^SSBP^    /'          tively  in    the    points  c   and  cf . 
^rS!  The  arcs  ab  and  a'b'  are  con- 
nected at  their  ends  by  small  cir- 
cle-arcs as  shown  in  the  figure. 
The  axes  of  pulley-arms  may  be  straight  as  in  Fig. 


ARMS  OF  PULLEYS. 


167 


58,  curved  as  in  Fig.  59,  or  double  curved  in  the  form 
of  a  letter  5.  Single-curved  arms  may  be  drawn  in  the 
following  manner:  Take  (Fig.  59)  the  arc  AE  equal  to 
•f  the  arc  EF,  determined  by  the  centre  s  of  the  arms  at 
the  rim  of  the  pulley,  and  draw  A^O  perpendicular  to 
AO.  From  the  centre  D  draw  CD  perpendicular  to 


Jl:' 


FIG.  58. 


FIG.  59. 


OE,  and  the  point  C  of  intersection  of  DC  and  OC  is 
the  centre  for  the  curved  axis  of  the  arm. 

The  number  of  arms  (TV)  necessary  for  pulleys  of 
different  sizes  may  be  determined  by  means  of  the 
formula 


or  the  following  table  calculated  from  it : 

R 

T-  =  i     2     3    4     5     6    7    8    9     10     11     12     13 


5 


1  68  BELTS  AND  PULLEYS. 

The  formula 


.    .     (610) 


gives  the  greater  diameter  for  the  pulley-arms.  The 
diameter  or  width  h  is  taken  at  the  nave  as  shown  in 
Fig.  58,  and  the  width  /il  at  the  rim  may  be  conven- 
iently taken  equal  to  J/z.  These  expressions  have  been 
determined,  with  a  certain  approximation  from  the 
most  accurate  formulas  ;  for  large  and  medium  sized 
pulleys  they  are  especially  applicable,  but  for  small 
light  pulleys  the  dimensions  should  be  slightly  in- 
creased in  order  that  the  pulleys  may  be  easily  cast 
without  taking  special  precautions. 

Example.  —  Required  the  numbef  of  arms  and  the 
arm  dimensions  for  a  pulley  having  a  radius  of  1  8  inches, 
the  belt  for  the  pulley  being  6  inches  wide.  Here 

R        i% 


From  the  above  table  we  find  the  number  of  arms  to 
be  N  =  4,  and  formula  (610)  gives  for  the  width  of  the 
arms  in  the  plane  of  the  pulley 

6  18 

h  =  0.24  +  -  +  -        -  =  2.19". 
1   4        10  X  4 

The  width  at  right  angles  with  the  plane  of  the  pulley 
is  therefore 

A,  =  |-  X  2.19=  1.46*. 


ARMS  OF  PULLEYS. 


169 


To  trace  the  profiles  of  the  arms  proceed  as  follows: 
Straight  arms  (Fig.  60). — Having  drawn  the  diameter 
EOC,  take  ab  •=•  cC  =  Cd  —  f //,  and  draw  the  lines  ac  and 
bd,  which  give  the  limits  of  the  profile.  Connect  ac  and 


FIG.  60. 


FIG.  61. 


bd  with  the  rim  and  nave  by  small  circle  arcs,  and  the 
profile  is  complete.  Curved  arms  (Fig.  61.) — The  centre 
C  for  the  axis  having  been  determined,  draw  the  straight 

line  ad,  then  take  aE  •=.  Eb  =  —  and  Cc  =  Cd  —  -? ;  the 

3  6 

points  c  and  d  thus  determined  are  the  centres  for  the 
arcs  which  limit  the  profile,  and  cb  and  da  are  the  radii. 
Double-curved  arms.* — Fig.  62  shows  a  simple 
method  for  drawing  double-curved  arms.  Draw  the 
radial  line  oA,  making  30°  with  the  horizontal.  Take 
oc  —  \oA,  and  through  the  point  c  draw  the  line  pD, 
making  60°  with  the  horizontal.  Intersect  the  line 


*From  the  author's   "Treatise  on  Toothed  Gearing." 


170 


BELTS  AND  PULLEYS. 


pD  by  a  horizontal  line  through  the  point  A  :  the  points 
D  and/  are  respectively  the  centres  for  the  arcs  oc  and 
cA,  which  together  form  the  axis  of  the  arm.  Lay  off 
the  arm-widths  as  shown  in  the  figure.  From  the 


FIG.  62. 

point  /  as  a  centre  strike  the  arcs  ab  and  ef,  and  find 
upon  the  line  oD  the  centres  for  the  remaining  arcs  bd 
and  fkf. 

Another  very  similar  method  for  drawing  double- 
curved  arms  is  shown  in  Fig.  63.     Draw  the  radial  line 

MM 


oA,  making  45°  with  the  horizontal.     Take  oc  =  -§oA, 
and  through  the  point  c  draw  the  vertical  line  pD. 


SHAFTS. 


171 


Intersect  the  line  pD  by  the  horizontal  line  Ap.  The 
points/  and  D  are  the  centres  for  the  arcs  of  the  axis. 
Lay  off  h  and  //,  as  shown  in  the  figure,  and  proceed, 
as  in  Fig.  62,  to  strike  the  arcs  ah,  efy  bd,  and  fk'. 

§15.    Shafts* 

When  a  shaft  is  so  supported  by  its  bearings  as  to 
be  subjected  to  a  torsional  strain  only,  as  is  almost  in- 
variably the  case  in  pulley-shafts  (the  bending  strain 
due  to  the  weight  of  the  pulley  and  the  force  trans- 
mitted by  the  belt  being  ordinarily  slight  enough  to  be 
safely  neglected),  the  calculation  of  the  proper  strength 
for  the  shaft  may  be  made  as  follows : 


FIG.  64. 


In  Fig.  64,  P  represents  the  total  force  tending  to 
twist  the  shaft,  i.e.,  the  total  force  transmitted  by  the 
belt ;  R  the  distance  from  the  centre  of  the  shaft  to 
the  point  at  which  the  force  acts,  i.e.,  the  radius  of  the 
pulley ;  and  d  the  diameter  of  the  pulley-shaft.  The 

*From  the  author's  "  Treatise  on  Toothed  Gearing." 


172  BELTS  AND  PULLEYS. 

greatest  safe  torsional  strain  which  can  be  sustained  by 
the  shaft  is  given  by  the  expression 


in  which  f  is  the  greatest  safe  shearing  stress  in 
pounds  per  square  inch  for  the  material  of  the  shaft. 
From  this, 


.        .  ,       PR 
d  — 


O.I9535/" 

or,  d=  1.720  \l-jr (6n) 

RULE. — To  determine  the  diameter  of  a  pulley-shaft 
of  any  material  multiply  the  total  force  transmitted  by 
the  belt  by  the  radius  of  the  pulley,  divide  this  pro- 
duct by  the  greatest  safe  shearing  stress  in  pounds  per 
square  inch  for  the  material  of  the  shaft,  extract  the 
cube  root  of  the  quotient  thus  obtained,  and  multiply 
the  result  by  1.720. 

Example. — Required  the  diameter  for  an  oak  shaft 
upon  which  is  a  6o-inch  pulley  transmitting  a  force  of 
loco  pounds,  taking  f  —  500  pounds.  From  formula 
(6 1 1)  we  have 


=  1.720  =  1.720x3-915  =  6.734"= 


We  propose  to  take  for  steel  ff  =  12000  pounds; 
for  wrought-iron/'  =  8000  pounds  ;  and  for  cast-iron 
f  =  4000  pounds.  These  values  of  f  are  nearly 


SHAFTS.  173 

mean  between  those  used  by  Stoney,  Haswell,  and 
Unwin,  which  differ  far  more  than  is  conducive  to  any 
degree  of  accuracy.  Substituting  the  above  values  of 
f  successively  in  formula  (611)  and  reducing,  we 
obtain, 

For  steel,  d  =  0.075  *V~PR (612) 

For  wrought-iron,  d  —  0.086  *V~PR (613) 

For  cast-iron,          d  =  o.io!&  *V~PR (614) 

RULE. — To  determine  the  diameter  for  a  pulley- 
shaft  of  steel,  wrought  or  cast  iron,  multiply  the  total 
force  transmitted  by  the  radius  of  the  pulley,  extract 
the  cube  root  of  the  product,  and  multiply  the  result 
by  0.075  for  steel,  0.086  for  wrought-iron,  and  0.108 
for  cast-iron. 

Example. — A  48-inch  pulley  transmits  a  force  of 
1000  pounds.  Required  the  diameter  for  a  steel  shaft. 
.From  formula  (612)  we  have 

d  —  0.075  VTooo  X  24  =  0.075  X  28.84, 
or,         d  —  2.163"  =  2ii/x  nearly. 

Example. — Taking  the  data  of  the  preceding  ex- 
ample, required  the  diameter  for  a  shaft  of  cast-iron. 
Formula  (614)  gives 


d  —  0.108  Viooo  X  24  =  0.108  X  28.84. 
or,         d  =  3- 1 1 5"  =  3-g-"  nearly. 

Formulas  for  the  diameters  of  pulley-shafts  in  terms 


174 


BELTS  AND  PULLEYS. 


of  the  horse-power  transmitted  and  the  revolutions  per 
minute  may  be  obtained  as  follows: 

As  before  explained,  we  have  the  expression 


P  =  63000 


Rif 


H  representing  the  horse-power,  R  the  radius  of  the 
pulley,  and  n  the  number  of  revolutions  per  minute. 
Substituting  this  value  in  formulas  (611),  (612),  (613), 
and  (614),  and  reducing,  we  obtain  the  following: 


General  formula,    d  =  68  44  \  /  — 


For  steel, 


For  wrought-iron,  d  =  3.422  A  / — .      ....     (617) 


d=  2.984^7- (616) 


For  cast-iron,          d  =  4.297 


H 


(618) 


RULE. — To  determine  the  diameter  for  a  pulley- 
shaft  of  any  material  from  the  horse-power  and  num- 
ber of  revolutions  per  minute,  divide  the  horse-power 
by  the  product  of  the  number  of  revolutions  into  the 
greatest  safe  shearing  stress  in  pounds  per  square  inch 
for  the  material  of  the  shaft,  extract  the  cube  root  of 
the  quotient  thus  obtained,  and  multiply  the  result  by 
68.44. 


SHAFTS.  175 

To  determine  the  diameter  for  a  pulley-shaft  of 
steel,  wrought  or  cast  iron,  from  the  horse-power  and 
number  of  revolutions  per  minute,  divide  the  horse- 
power by  the  number  of  revolutions,  extract  the  cube 
root  of  the  quotient,  and  multiply  the  result  by  2.984 
for  steel,  3.422  for  wrought-iron,  and  4.297  for  cast-iron. 

Example.  —  Required  the  diameter  for  an  oak  pulley- 
shaft  which  transmits  a  force  of  10  horse-power  and 
makes  40  revolutions  per  minute.  If  we  take  for  the 
greatest  safe  shearing  stress  for  oak  f  —  500  pounds 
per  square  inch,  we  shall  have,  from  formula  (615), 


~-^-> 
12.60 


3     /  IO  3/1 

dfr=  68.44  \     —      -—  68.44A/-     -  =  68. 
vy  40x500  fV  2000 

or,  d  =  5.432"  =  5Ty  nearly. 

Example.  —  Taking  the  data  of  the  preceding  ex- 
ample, required  the  diameters  for  shafts  of  steel  and 
wrought-iron. 

From  formula  (616). 

d  =  2.984  \     —  —  2.984  Vo.25  —  2.984  X  0.62996, 
V  4° 

or,  for  steel,  d  =  1.88"  =  i£|". 

From  formula  (617), 

3     /IO 

d  ==  3422  W  —  =  3422  X  0.62990, 
or,  for  wrought-iron, 


BELTS  AND  PULLEYS. 


Pulley-shafts  are  most  commonly  of  wrought-iron ; 
when,  however,  wrought-iron  shafts,  in  order  to  give 
the  necessary  strength,  become  so  large  as  to  be  incon- 
venient, steel  shafts  are  used.  Cast-iron  shafts  are,  as 
a  rule,  unreliable  and  treacherous  ;  they  are  therefore 
seldom  used  except  for  the  transmission  of  slight 
powers  and  in  cheap,  inferior  machinery.  The  follow- 
ing tables,  calculated  from  formulas  (612),  (613),  (616), 
and  (617)  to  the  nearest  -^  inch,  will  be  found  very 
convenient  in  designing  pulley-shafts  of  steel  and 
wrought-iron : 

TABLE  OF  SHAFT-DIAMETERS. 


d  for  steel. 


rtf  for 
wrought-iron. 


PR 


d  for  steel. 


dior 
wrought-iron. 


250 

500 
1000 

1500 

2000 
25OO 
3000 
3500 
4OOO 
4500 
5OOO 
6000 
7000 
8000 
1 0000 

12500 
15000 

20000 

25  oo 
30000 
35000 

40000 

45000 
50000 


'A 
'H 


21% 

2M 


60000 
70000 
80000 
90000 

I 00000 
IIOOOO 
120000 

130000 
140000 
150000 
175000 

200000 

250000 
500000 
750000 

I 000000 

1500000 

2000COO 

2500000 
3000000 
3500000 
4000000 
4500000 
5000000 


SHAFTS. 


TABLE  OF  SHAFT-DIAMETERS. 


H 


d  for  steel. 


^  for 
wrought-iron. 


H 


d  for  steel. 


dior 
wrought-iron. 


0.025 
0.050 
0.075 
O.IOO 

0.150 

0.200 
O.25O 
0.300 
0.350 
O.4OO 

o  500 
0.600 
0.700 
0.800 
0.900 
I. 

1.25 
1.50 

1-75 

2. 

2.25 

2.50 

2.75 

3- 

3-25 

3-50 


If 


2i 
2| 
2| 
2| 
2| 


3|| 
4<ftr 
4H 
4A 
4» 


3-75 

4 

4-25 

4.50 

4-75 

5 

5-50 

6 

6.50 

7 
8 

9 
10 
ii 

12 
-14 

16 

18 

20 
22 

25 
27 
30 
32 

35 
40 


Example. — Required  the  diameter  for  a  wrought- 
iron  shaft  for  a  4O-inch  pulley  which  transmits  a  force 
of  1000  pounds.  In  this  case 

PR  —  1000  X  20  =  20000, 

and  from  the  table  on  page  176,  the  value  of  d  for 
wrought-iron  corresponding  to  PR  =  20000  is  d  —  2f^ 
inches. 

Example. — The  diameter  of  a  wrought-iron  pulley- 

12 


178  BELTS  AND  PULLEYS. 

shaft  is  4j  inches.  Required  the  force  which  the  shaft 
can  safely  transmit  by  means  of  a  24-inch  pulley.  From 
the  table  on  page  176  the  value  of  PR  corresponding 
to  d  =  4|-  inches  for  wrought-iron  is  110,000;  hence 
we  have 

IIOOOO         IIOOOO 

/*=  — 75 —  = =  9107  pounds  nearly. 

.A.  1 2 

Example. — A  pulley  transmitting  a  force  of  20  horse- 
power makes  200  revolutions  per  minute.  Required 
the  diameter  for  a  shaft  of  steel.  We  have 

H       20        i 
-  =  —  —  —  =  0.100, 
n       200      10 


and  from  the  table  on  page   177  the  value   of  d  for 

rr 

steel  corresponding  to  —  =  o.ioo  is  d  =.  iff  inches. 

72 

Example. — A  2-inch  steel  shaft  transmits  a  force  of 
25  horse-power.  It  is  required  to  determine  the 
proper  number  of  revolutions  per  minute.  From  the 

IT 

table  on  page  177  the  value  of  —  which  corresponds  to 

TT 

d  =  2  inches  for  steel  is  —  =  0.300 ;  hence  we  have 


H     25 

-==—==  0.300, 
n        n          ° 


or  n  —  83^  revolutions  per  minute. 


THE    TIGHTENING-PULLEY. 


179 


§  1 6.     The   Tighten ing-Pu//ey.— Fast  and  Loose  Pulleys. 

Tightening-pulleys  are  used  to  tighten  loose  belts, 
or,  in  other  words,  to  increase  the  tension,  and  thus 
prevent  slipping  upon  the  principal  pulleys.  Fig.  65 
represents  a  tightening-pulley  as  commonly  used  in 
the  shops.  A  and  B  are  the  principal  pulleys,  and  C 
the  tightening-pulley,  which  is  pressed  against  the  belt 


FIG.  6e;. 


or  raised  off  the  belt  by  means  of  the  lever  d.  Often 
the  weight  of  the  tightening-pulley  is  sufficient  to  pro- 
duce the  required  tension  ;  if  not,  extra  weights  are 
hung  to  the  pulley,  or  the  lever  fastened  up  in  its  proper 
position.  When  the  pulley  C  is  lifted  off  the  belt  en- 
tirely, the  belt  relieved  of  its  tension  no  longer  runs 
upon  the  driver ;  the  driven  pulley  is  then  at  rest,  or 
the  belt  is  disengaged.  The  tightening-pulley  obviates 
the  necessity  of  taking  up  the  slack  caused  by  the 
stretching  of  the  belt,  for  as  the  belt  becomes  longer 


180  BELTS  AND  PULLEYS. 

and  consequently  looser  upon  the  principal  pulleys  by 
stretching,  it  may  be  tightened  by  simply  lowering  the 
tightening-pulley.  A  glance  at  the  figure  will  show 
that  by  means  of  this  pulley  the  arcs  of  contact  be- 
tween the  belt  and  the  principal  pulleys  are  increased 
to  a  considerable  extent — which  in  itself  is  an  impor- 
tant consideration.  The  tightening-pulley  is  also  a 
valuable  means  of  increasing  the  duration  of  a  belt ; 
for  since  the  wear  upon  the  belt  increases  with  the  ten- 
sion to  which  it  is  subjected,  it  is  important  that  the 
tension  be  no  greater  than  is  sufficient  to  prevent  slip- 
ping, and  this  may  be  easily  regulated  by  lifting  or 
lowering  the  lever  which  controls  the  position  of  the 
pulley.  By  placing  the  tightening-pulley  below  the 
belt  the  contrivance  may  also  be  made  to  take  the 
place  of  a  pulley  support.  With  high-speed  belts  con- 
siderable care  is  necessary  to  keep  the  tightening-pulley 
in  its  proper  position. 

Fast  and  loose  pulleys  are  used  as  a  means  of  en- 
gaging and  disengaging  the  belt,  and  thus  starting  or 
stopping  the  driven  pulley  without  interfering  in  any 
way  with  the  driver.  This  is  a  very  necessary  con- 
sideration in  cases  where  several  machines  are  driven 
by  a  single  driving-pulley,  as  is  almost  always  the  case 
in  practice.  Many  contrivances  have  been  from  time 
to  time  devised  for  this  purpose,  but  few  if  any  have 
proved  as  simple  and  sure  as  the  fast  and  loose  pulleys 
seen  in  nearly  every  shop  and  factory  in  the  land. 
Fig.  66  represents  a  pair  of  such  pulleys.  A  is  keyed 
fast  to  the  shaft  C  C,  while  the  pulley  B  runs  loose 
upon  the  shaft.  The  belt  is  made  to  pass  from  one 
pulley  to  the  other  by  means  of  a  lever,  or  similar  device. 


THE    TIGHTENING-PULLEY. 


181 


When  the  belt  is  on  the  fast  pulley  A,  the  motion 
of  the  driving-pulley  is  transmitted  to  the  shaft  C  C. 
When  the  belt  is  on  the  loose  or  idle  pulley  B,  this 
pulley  simply  rotates  upon  the  shaft  without  giving  to 
it  any  motion.  In  many  cases  the  loose  pulley  is  placed 
upon  the  driven  shaft ;  the  belt  then  continues  its  mo- 
tion when  upon  the  loose  pulley.  It  is  preferable, 
however,  to  have  the  loose  pulley  on  the  driving-shaft, 
because  when  the  belt  is  out  of  gear  it  remains  mo- 


FIG.  66. 


,  tionless,  thus  saving  it  from  unnecessary  wear.  Often 
there  are  two  loose  pulleys,  one  on  each  shaft  ;  the 
driving  fast  pulleys  are  then  of  the  same  face-width, 

*  while  with  one  loose  pulley  the  driving-pulley  on  the 
other  shaft  must  have  a  face-width  equal  to  those  of  the 
loose  pulley  and  its  neighboring  fast  pulley  together. 

A  device  introduced   some  years  ago   for  the   pur- 
pose  of  diminishing  the    tension    upon    a   belt   while 


182 


BELTS  AND  PULLEYS. 


upon  a  loose  pulley  is  shown  in  Fig.  67.  A  is  an  or- 
dinary pulley  keyed  fast  to  the  shaft  D  D,  and  B  a 
loose  pulley,  which  is  somewhat  smaller  than  the  fast 
pulley,  and  which  carries  a  conical  flange  C  C,  the  out- 
side diameter  of  which  is  equal  to  that  of  the  fast  pul- 
ley A.  When  the  belt  passes  from  A  to  B,  the  tension 


FIG.  67. 


upon  it  is  diminished,  the  belt  slackens,  and  while  out 
of  work  is  not  subjected  to  any  considerable  strain. 
In  ordinary  fast  and  loose  pulleys  the  tension  upon  the 
belt  is  constant,  whether  the  belt  is  at  work  or  at  rest. 
Fig.  68  represents  a  common  application  of  the 
principle  of  fast  and  loose  pulleys,  by  which  an  alter- 
nate rotating  motion  in  both  directions  is  obtained 
from  the  continuous  rotary  motion  of  the  driving-shaft. 
The  pulleys  A,  C,  A',  and  C  are  fast,  while  B  and  B' 
run  loosely  upon  their  shafts.  Two  belts,  one  open 
and  the  other  crossed,  are  placed  side  by  side  in  such 
a  manner  that  one  rests  upon  the  loose  pulleys,  while 
•the  other  runs  upon  one  or  the  other  pair  of  fast  pul- 


THE    TIGffTEWlNG-PULLEY. 


183 


leys.  When  the  belts  are  in  the  positions  shown  in 
the  figure,  the  crossed  belt  is  the  driver,  and  the  open 
belt  remains  motionless.  By  sliding  the  two  belts 
over  the  pulley-faces  the  open  belt  is  placed  upon  the 
fast  pulleys  A  and  A\  and  the  crossed  belt  upon  the 
two  loose  pulleys.  This  reverses  the  direction  of 


FIG.  68. 

rotation  of  the  driven  shaft,  and  by  sliding  the  belts 
back  into  their  first  positions  the  motion  of  the  driven 
shaft  is  again  reversed. 

The  most  familiar  example  of  this  reversing  gear  is 
seen  in  the  planing-machine,  where  the  forward  and 
backward  motion  of  the  table  which  carries  the  work 
is  thus  accomplished. 

Belts  with  circular  cross-sections,  such  as  round 
leather-belts,  rope-belts,  etc.,  generally  have  pulleys 
with  grooved  faces.  The  ordinary  fast  and  loose  pul- 


1 84 


BELTS  AND  PULLEYS. 


leys  obviously  cannot  be  used  in  such  cases.  Fig.  69 
shows  a  fast  and  loose  pulley  for  round  belts,  which 
seems  to  answer  the  purpose  very  well.  The  part  B 
of  the  fast  pulley  is  keyed  fast  to  the  shaft  dd,  and 
the  part  A  may  be  moved  away  from  the  part  B  by 
means  of  the  lever  f.  In  the  figure  the  parts  of  the 


FIG.  69. 


fast  pulley  are  together,  and  the  belt  gg  therefore 
drives  the  shaft.  When  the  parts  are  separated  the 
belt  slides  from  the  part  B  to  the  inside  loose  pulley 
C,  which  then  rotates  about  the  shaft  without  trans- 
mitting to  it  its  motion.  Upon  sliding  the  part  A 


ROPE-BELTS. 


135 


again  into  the  position  shown  in  the  figure  the  thin 
rim  slides  under  the  belt  and 
lifts    it    into    the  groove,    in 
position  for  work. 

Another  fast  and  loose 
pulley  for  round  belts  is 
represented  in  Fig.  70.  In 
this  case  the  pulley  runs 
loosely  upon  the  shaft  dd 
when  in  the  position  shown 
in  the  figure.  The  conical 
key  B  is  fast  to  the  shaft, 
and,  when  forced  into  the 
hub  of  the  pulley  by  means 
of  the  lever  /",  bites  with  suf- 
ficient force  to  secure  the 
pulley.  A  collar  kk  fast  to 
the  shaft  prevents  the  pul- 
ley from  sliding  away  from 
the  key.  For  light  transmissions  this  pulley  may  work 
satisfactorily,  but  for  heavy  or  unsteady  work  it  can 
hardly  be  spoken  of  as  reliable. 


ff 


FIG.  70. 


§17.     Rope-Belts. 

Hemp  and  cotton  ropes  are  sometimes  used  for 
transmission-belts,  the  principal  pulleys  being  placed 
from  25  to  60  feet  apart.  Three-strand  ropes,  such  as 
is  represented  in  section  in  Fig.  71,  are  most  commonly 
used  ;  the  diameters  vary  from  f  inch  to  2j  inches, 
and  by  placing  several  ropes  side  by  side  upon  the 
principal  pulleys,  large  powers  may  be  transmitted. 


1 86 


BELTS  AND  PULLEYS. 


As  a  general  rule,  rope-belts  work  almost  entirely  by 
means  of  their  weights,  being  hung  loosely  upon  the 
pulleys  instead  of  tightly  stretched  over  the  pulleys  as 
in  leather  and  vulcanized-rubber  belts.  When  only 
small  powers  are  to  be  transmitted,  the  pulleys  may 
have  semicircular  grooves  upon  their  faces,  and  the 
rope-belts  may  run  in  the  bottoms  of  the  grooves. 
The  weight  of  the  belt  in  such  cases  furnishes  sufficient 
friction  to  prevent  slipping.  Fig.  72  represents  a  pul- 
ley-rim of  this  kind  for  a  single  rope-belt.  When, 


FIG.  71. 


FIG.  72 


however,  large  powers  are  to  be  transmitted,  the  grooves 
in  the  pulley-faces  should  be  V-shaped,  so  that  the 
ropes  may  be  wedged  between  the  sides,  and  thus  fur- 
nish the  friction  lost  by  diminishing  the  initial  tension. 
A  pulley-rim  for  four  rope-belts  transmitting  large 
power  is  represented  in  Fig.  73.  The  dimensions  of 
the  pulley  may  be  calculated  as  for  ordinary  pulleys, 
and  the  sides  of  the  grooves  commonly  make  angles  of 
45°  with  each  other,  as  shown  in  the  figure. 

The  coefficient  of  friction  for  rope-belts  running  in 
the  bottoms  of   semicircular  grooves  without   biting 


ROPE-BELTS. 


187 


against  the  sides  (Fig.  72)  may  be  taken,  for  cast-iron 
pulleys, 

<p  =  0.30. 

For  V  grooves,  of   which  the    sides  are   inclined   at 
angles  of  45°,  as  in  Fig.  73, 

cp  —  0.70 (619) 


FIG.  73. 

By  substituting  this  value  in  formula  (41),  we  obtain  for 
the  ratio  of  the  tensions 

T 

log--  =  0.005  3  tf,    ....    (620) 

in  which  a  is  expressed  in  degrees. 


If  in  formula  (48)  we  make  the  quantity 


T 

t 


~r 


we  have 


T=  Px. 


(621) 


1 88 


BELTS  AND  PULLEYS. 


T 


The  following  table  gives  values  for  --  and  x  for  all 
values  of  a  likely  to  be  needed  in  practice : 


a  in 

degrees. 

a  in  circular 
measure. 

a  in  fractions  of 
circumference. 

T 

t     • 

X. 

45 

0.785 

i  =  0.125 

1.732 

2-37 

60 

1.047 

\  —  0.167 

2.080 

T.Q3 

75 

1.309 

•ff  =  0.208 

2.498 

1.67 

90 

J-57I 

J  =:  O.25O 

3.001 

•50 

105 

1.833 

A  =  0.292 

3.602 

•38 

1  20 

2.094 

i  =  0-333 

4.325 

-30 

135 

2.356 

1  =  0.375 

5-194 

.24 

1  80 

3.142 

-£  =  0.500 

8.995 

.12 

210 

3-665 

A  =  0.583 

12.97 

.08 

240 

4.188 

1  =  0.667 

18.71 

.06 

If  we  represent  the  diameter  of  the  rope  by  6',  we 
have  for  the  area  of  cross-section 


=  0.785*", 


and  this  substituted  for  b$  in  formula  (50)  gives 

0.785*"  =   - 


The  safe  working  stress  in  pounds  per  square  inch  may 
be  taken 

/=  1200, 
but  for  greatest  durability  and  best   performance  of 


ROPE-BELTS.  189 

work   this   is   in   practice  about  TV  the   above  value. 
Hence  we  use 

/=    120, 

which  substituted  in  the  above  formula  gives 


or  =  o.i0 

and  consequently 

&  =  0.103  tfPx  .....     (622) 
As  before  explained,  we  have  the  expression 


in  which  H  represents  the  horse-power  transmitted  and 
v  the  velocity  in  feet  per  second.  By  substituting  this 
in  formula  (622)  we  obtain 


I H 
or  V  =  2.416^1 —x (623) 


Example. — Required  the  proper  diameter  for  a  rope- 
belt  which  will  transmit  a  force  of  1000  pounds  over 


BELTS  AND  PULLEYS. 

two  equal  V-grooved  pulleys.     In  this  case  a  =  180°, 
and  the  table  gives 

x  =  1. 12. 
Hence,  from  formula  (622), 


d'  —  0.103  Viooo  X  1. 12  =  0.103  X  3347, 
or  tf'  -  3447'  -  3ff  • 

Rope-belts  as  large  as  this  are  seldom  used  in  prac- 
tice. In  the  above  example,  therefore,  we  should  use 
two  ropes  instead  of  one.  Each  rope  would  then 

transmit  a  force  of  -     -*••==  500  pounds,  and  we  should 


have  6'  =  0.103  1/500  X  1.12, 

or  rf'  =  2.437"  =  %Y'- 

Example.  —  A  rope-belt  embracing  an  angle  of  135° 
upon  its  smaller  principal  pulley  transmits  a  force  of 
15  horse-power  at  a  velocity  of  30  feet  per  second.  It 
is  required  to  determine  the  proper  diameter  for  the 
belt.  From  the  table  we  have 

x  =  1.24, 
and  from  formula  (623) 


'  =  2.416 


or  8'  =  1.901"  =  iff". 


ROPE-BELTS.  igi 

Example.—  It  is  required  to  transmit  a  force  of  800 
horse-power  at  a  velocity  of  80  feet  per  second  by 
means  of  15  rope-belts.  The  arc  embraced  by  each 
belt  upon  the  smaller  principal  pulley  is  equal  to  f  the 
circumference.  Required  the  diameters  for  the  rope- 
belts. 

It  is  evident  that  each  belt  must  transmit  a  force  of 


—  50  horse-power  at  a  velocity  of   80   feet   per 

second. 

The  table  gives,  for  a  =  f  the  circumference, 

x  =  1.24, 
and  from  formula  (623)  we  have 


*'  -  2.416  y _        =  24l6  x  0>88> 

or         6'       2.126"  =  2i". 

Because  of  the  circular  cross-sections  of  rope-belts 
and  the  character  of  the  material  generally  used,  it  is 
necessary  that  the  wear  due  to  the  bending  of  the 
ropes  about  the  pulleys  be  reduced  as  low  as  possible. 
To  this  end  very  large  principal  pulleys  are  used — from 
about  7  to  15  feet  in  diameter  commonly.  It  is  a  safe 
rule,  that  the  diameter  of  the  smaller  principal  pulley 
should  not  be  less  than  thirty  times  the  diameter  of 
the  rope,  and  when  small  ropes  are  used  we  may  con- 
veniently increase  the  durability  by  taking  the  diameter 
of  the  smaller  pulley  equal  to  45  to  60  times  the  diame- 
ter of  the  rope.  Thus  in  the  three  examples  given 


IQ2  BELTS  AND  PULLEYS. 

above  we  may  have  for  the  diameters  of  the  smaller 
principal  pulleys  30  X  2.44  =  73.20''  =  6J  feet,  30  X 
i-9  =  57"  =  5  ^et  nearly,  and  30  X  2.13  =  63.9"  = 
Si  feet. 

The  ends  of  rope-belts  are  usually  spliced  together 
by  pressing  them  firmly  together  and  winding  about 
with  stout  small  rope.  The  spliced  part  should  be  as 
long  as  possible  in  order  to  bend  properly  over  the 
pulleys  and  give  the  necessary  strength.  The  weight 
per  foot  of  length  of  rope-belts  is  approximately  given 
by  the  formula 

G  =  o.3<P (624) 

§  1 8.     Jointed  Chain-Belts.* 

Of  late  years  numerous  attempts  have  been  made 
to  replace  ordinary  leather  belts  by  traction-bands. 
Among  the  various  systems  proposed  we  mention  in 
the  first  place  the  chain-belt  (leather)  of  Rouiller :  this 
contrivance,  which  at  first  appeared  destined  to  do 
good  service,  has  not  justified  this  hope,  but  has  fallen 
into  disuse  because  of  its  want  of  durability.  Belts 
formed  of  twisted  metallic  wires  (Godin)  have  produced 
results  scarcely  more  satisfactory.  As  for  leather  belts 
covered  with  gutta-percha,  they  cannot,  in  reality,  com- 
pete with  ordinary  leather  belts ;  and  at  the  present 
time  there  is  scarcely  a  transmission-band,  with  the  ex- 
ception of  rubber-belts  with  layers  of  hemp  or  cotton, 
which  seems  to  be  as  advantageous  in  practice  as  or- 
dinary leather  belts,  especially  when  used  for  the  trans- 
mission of  considerable  forces. 

*  From  Reuleaux. 


JOINTED   CHAIN-BELTS.  193 

In  certain  special  cases,  for  the  transmission  of  large 
forces,  and  for  unsteady  work,  such  as  in  agricultural 
machines,  the  ordinary  leather  belt  may  be  successfully 
replaced  by  the  jointed  chain-belt  of  Clissold  (Fig.  74). 
In  this  chain  the  joints  are  bound  together,  two  by 
two,  by  leather  bands  wound  several  times  around,  and 
bevelled  at  the  edges  to  fit  properly  in  the  trapezoidal 
groove  which  forms  the  face  of  the  pulley.  Angstrom 


FIG.  74. 


has  used  instead  of  the  leather  bands  pieces  of  wood 
trimmed  with  iron. 

In  calculating  the  tensions  for  jointed  chain-belts  it 
is  necessary  to  introduce  the  friction  of  the  joints  in- 
stead of  the  rigidity  which  figures  in  formulas  for 
leather-belts.  The  formulas  for  tensions  of  leather- 
belts  may  be  used  in  the  present  case  by  putting 

cpa  = p ,  0  representing  the  angle  of  the  bevelled 

edges  of  the  chain-belt. 


For  cp  =  0.24,  a  —  O.STT, 

i  -» 


=  30°,  we  obtain 


IQ4  BELTS  AND  PULLEYS. 

t  T  T+t  t 

p  =  0.20,    -p=  1.23,    — p—  =  1.43,    j.=  0.163  ;    (625) 

and  for  q>  =  0.28,  a  =  0.95^,  6)  —  30°, 
t  T  T  I  /  / 

^  =  0.12,     ^=1.15,      -±-=1.27,     ^,  =  0.105.      (626) 

By  making  use  of  these  values  we  may  obtain  for 
the  diameter  of  the  joint  pivots  (d.  Fig.  74)  the  ex- 


~ff 

d  =  0.0146  VP=  3.656  ;    ....    (627) 


rrr  0.04H  -(PR)  =  1.644      /T  ~.  •       •      (628) 


We  should  take  for  jointed  chain-belts  the  following 
proportions  (see  Fig.  74)  : 

/  b  c        i      e       i     h 

d  =  ^    d  =  *'  d  =  i>   d  =  s'  d  =  2*'    (629) 

For  small  pulleys  it  is  convenient  to  take 


In  practice  d  should  not  be  taken  less  than  0.32  inch, 
even  when  a  smaller  diameter  would  be  sufficient  for 
strength.  In  jointed  chain-belts  the  limit  of  the  force 

*  P  =  force  in  pounds  transmitted,  //  =  horse-power,  n  ~  revolu- 
tions pc*'  minute. 


JOINTED   CHAIN-BELTS.  195 

P  which  maybe  transmitted  (supposed  to  be  applied 
at  the  circumference  of  the  pulley)  is  about  500  pounds, 
which  would  require  a  width  cf  about  1 1  inches  in  a 
simple  leather  belt. 

Example. — Given  the  data  H  =  20,  n  =  50,  n,  =  100. 
Required  the  dimensions  for  a  jointed  chain-belt,  sup- 
posing the  radius  of  the  smaller  pulley  to  be  Rt  =  5/. 
Formula  (628)  gives 


.  ,  20 

d-.-.  1.644      X  — 


=  i.644  \     --  =  -      ±  =  0.5624"  =  A". 
V  25       2.924 

From  formula  (629)  then  we  obtain  /  =  3  X  0.5624  = 
1.6872",  b  =  2|  X  0.5624  =  i  5466",  c  =  0.2",  *  = 
0.12",  h  —  2\  X  0.5624  =  1.2185",  -ffj  =  s/  =  8.436", 
R  =  2XR,=  16.872". 

Clissold  has  also  invented  a  transmission  by  means 
of  a  thick  belt  with  trapezoidal  section.  This,  how- 
ever, has  proved  poorly  because  of  its  want  of  dura- 
bility.* 

*  The  experiments  of  Wedding  of  Berlin  have  shown  that  in  an 
Angular  groove,  the  angle  being  30°  (Fig.  50),  the  force  necessary  to 
produce  slipping  of  the  cable  is  twice  that  corresponding  to  a  cable 
lying  in  a  round  groove.  This  confirms  the  preceding  expressions, 

i  i 

since 


sin  30         2 


TRANSMISSION  BY  METALLIC  CABLE. 


§  19.     Tensions  of  Cables. 

Transmission  of  forces  by  means  of  metallic  cables 
was  first  introduced  about  the  year  1850,  by  the  Hirn 
brothers.f  The  use  of  metallic  cables,  by  means  of 
which  we  are  able  to  transmit  great  forces  at  distances 
as  great  as  several  thousand  feet  without  notable  loss, 
depends  essentially  upon  the  principles  of  transmission 
by  belt,  the  principal  difference  being  that  with  a 
metallic  cable  the  tension  is  due  to  its  own  weight. 

The  two  principal  pulleys  of  a  transmission  by  cable, 
as  a  general  thing,  have  their  axes  parallel ;  also  the 
pulleys  are  in  the  same  plane,  so  that  the  cable  may 
be  driven  without  guides.  Moreover,  the  axes  of  the 
principal  pulleys  are  ordinarily  in  the  same  horizontal 
plane,  forming  what  is  termed  a  horizontal  transmis- 
sion. An  inclination  of  the  plane  of  the  axes  to  the 

*  From  Reuleaux. 

f  In  this  first  application  the  axes  of  the  pulleys  were  about  280  feet 
apart;  the  force  transmitted  was  42  horse-power,  at  60  revolutions  per 
minute. 


TENSIONS  OF  CABLES.  197 

surface  of  the  ground  constitutes  an  oblique  transmis- 
sion. Vertical  transmissions  by  metallic  cable  are  very 
rarely  used.  When  the  driven  pulley  transmits  to  a 
third  pulley  the  force  which  it  receives  from  the  driver 
the  transmission  is  said  to  be  compound.  In  a  simple 
transmission  by  cable  the  two  pulleys  are  ordinarily  of 
the  same  diameter. 

In  order  to  prevent  the  cable  from  touching  the 
ground,  when  the  height  of  the  pulleys  above  the 
ground  is  insufficient  and  the  separation  of  the  axes 
great,  intermediate  rollers  are  used  to  support  the 
cable.  By  inclining  the  rollers  more  or  less  they  may 
be  used  for  guides  when  the  axes  of  the  pulleys  cross 
or  intersect  each  other.  We  meet,  however,  very  few 
examples  of  transmission  by  cable  in  which  the  axes 
of  the  pulleys  are  not  parallel.  When  it  becomes 
necessary  to  give  to  the  cable  a  considerable  deviation, 
we  can  place  between  two  vertical  rollers  a  horizontal 
guide ;  but  it  is  preferable  in  such  cases  to  rely  upon  a 
compound  transmission,  with  pulleys  placed  obliquely 
to  each  other. 

The  inferior  limit  for  the  separation  of  the  pulley- 
axes  in  transmissions  by  metallic  cable  should  be  about 
50  feet. 

,  The  distances  between  the  rollers  which  support  the 
cable  are  determined  by  the  flexibility  of  the  cable 
and  its  position  above  the  ground. 

The  transmission-cables  ordinarily  used  are  com- 
posed of  36  iron  wires  divided  into  six  twists,  each 
containing  six  wires  twisted  around  a  central  core  of 
hemp ;  the  six  twists  are  likewise  twisted  around  a 
larger  core,  also  of  hemp  (Fig.  75).  When  it  is  neces- 


198 


BELTS  AND  PULLEYS. 


sary  to  strengthen  the  cable,  we  may,  without  serious 
disadvantage,  replace  the  central  hempen  core  by  a  twist 
of  iron  wire  similar  to  the  six  others.  It  has  also  been 
proposed  to  replace  by  an  iron  wire  the  smaller 
hempen  cores  of  the  separate  twists,  in  order  to  over- 
come the  looseness  of  the  cable,  which  may  tend  to 
produce  a  rapid  wear.  The  value  of  such  an  arrange- 
ment yet  remains  to  be  established.  It  has  the  dis- 


FIG.  75. 


FIG.  76. 


advantage  of  destroying  the  elasticity  of  the  cable. 
When  the  cores  are  of  hemp,  it  is  of  first  importance 
that  first  quality  hemp  be  used  in  their  manufacture, 
instead  of  the  inferior  qualities  which  have  been  hither- 
to extensively  used  for  this  purpose.  The  wires  com- 
posing the  cable  should  be  forced  firmly  together,  so 
that  the  diameter  of  the  cable  is  not  more  than  eight 
times  that  of  the  wire. 

In  cables  having  more  than  36  wires  the  number  of 
twists  is  generally  six,  and  the  large  and  small  cores  of 
hemp. 


TENSIONS  OF  CABLES.  199 


f 


While  there  is  no  absolute  necessity  of  limiting  the 
number  of  twists  to  six,  this  number  is  almost  always 
used:  in  the  different  cables  in  use  the  total  number 
of  wires  is  therefore  36,  48,  54,  60,  66,  72,  etc. 

Fig.  76  represents  a  cross-section  of  a  cable  of  60 
wires.  In  these  different  cables  the  relations  between 
the  external  diameter  d  and  the  diameter  d  of  the 
wires  are  as  follows  : 

For  the  number  of  wires  — 

i  =    36  48          54          60          66          72, 

-  —  8.00       10.25      11.33      12.80     13.25      14.20. 

In  order  to  obtain  the  tensions  T  and  t  in  metallic 
cables  we  make  use  of  the  formulas  determined  for 
tensions  in  ordinary  belts.  By  substituting  in  these 
formulas  a  coefficient  of  friction  cp  —  0.24,  and  an  arc 
of  the  pulleys  equal  to  -J-  the  circumstance,  a  —  iSo° 
=  n,  we  may  obtain  the  relations 

T  T+t  t 

=  z-02,  —   -  =  2.99,      ,=  0.48;    (630) 


'  or,  in  round  numbers, 


T  T+t  t 

>  =  2,   —r-  =3,       =0.5.*   .    .    (631) 


*  The  loss  of  velocity  due  to  the  shipping  of  the  cable  does  not 
ordinarily  exceed  -fa  per  cent;  it  may  therefore  be  neglected  alto- 
gether in  our  calculations. 


20O  BELTS  AND  PULLEYS. 

§  20.  Calculation  of  Diameters  of  Cables. 
In  a  transmission  by  metallic  cable  composed  of  i 
wires  the  tension  T  in  the  cable  corresponds  to  a  ten- 
sion 5  in  the  wires;  this  tension  should  not  exceed 
25601.4  pounds  per  square  inch  of  section.*  To  de- 
termine the  diameter  d  of  the  wires  the  following  for- 
mulas may  be  used : 

For  a  resistance  of  P  pounds  acting  at  the  circum- 
ference of  the  pulley, 

fP 

--  I-627y^ (632) 

For  a  force  of  H  horse-power,  with  a  velocity  of  v 
feet  per  second  at  the  circumference  of  the  pulley, 


7^%,  . 


;37-867y<-,.     •     •     . 

in  which  v  should  not  materially  exceed   100  feet  per 
second. 

For  a  force  of  H  horse-power  at  n  revolutions  of  the 
pulleys  per  minute, 

*  =  4070.04^/5^.    .  .  (634) 

If  we  represent  by  s  =  25601.4  —  5  the  tension  pro- 
duced in  the  wires  by  the  bending  of  the  cable  around 
the  pulleys,  and  by  (PR)  the  statical  moment  of  rota- 
tion of  the  driven  pulley,  we  shall  have 


(635) 


*  18  kilograms  per  square  millimetre. 


CALCULATION  OF  DIAMETERS  OF  CABLES.      2OI 


Finally,  if  in  place  of  the  moment  (PR)  we  have  the 
horse-power  and  revolutions  per  minute, 


d  =  0.227 


H 


Sni 


• (636) 


It  is,  moreover,  important  that  the  ratio  of  the  radius 
of  the  pulleys  to  the  diameter  of  the  wires  be  taken 
not  less  than  the  limit, 


R  _  14223000 


(637) 


This  relation  serves  to  calculate  the  following  table : 


s 

s 

R 
8 

s 

J 

R 
8 

711.15 

24890.25 

571 

12800.70 

12800.70 

mi 

1422.30 

24179.  10 

588 

14223.00 

11378.40 

1250 

2844.60 

22756.80 

625 

15645.30 

9956.10 

1429 

4266  .  go 

21334.50 

667 

17067.60 

8533.80 

1667 

5689.20 

19912.20 

714 

18489.90 

7111.50 

2000 

7IH.50 

18489.90 

769 

19912.20 

5689.20 

2500 

8533.80 

17067.60 

833 

21334.50 

4266.90 

3333 

9956.10 

15645.30 

909 

22756.80 

2844.60 

5000 

11378.40 

14223.00 

1000 

24179.10 

1422.30 

ICOOO 

For  a  constant  value  of  S  -\-  s  the  minimum  value  of 
the  radius   of  the   pulleys   is  given  by  the  table   by 

making  ~  =  2.*     This  minimum  value  corresponds  to 

o 

8/T~ 

*  We   may   obtain    from   formulas   (636)  and   (637)   R  =  Ky  -— 

'         S    O 

The  sum  s  -{-  S  being  constant,  the  maximum  value  of  the  product 
obtained  by  making  —  =  2. 


202 


BELTS  AND  PULLEYS. 


/?  7? 

S=  8533.8,  s  =  17067.6,  £  —  833.      For  values  of  y 

nearly  equal  to  833  the  numerical  value  of  R  differs 
very  little  from  the  minimum  value  ;  we  may  there- 
fore safely  give  somewhat  greater  values  to  R  when, 
by  so  doing,  we  can  make  use  of  patterns  and  models 
already  on  hand. 

The  two   tables  which  follow  have  been  calculated 
from  formulas  (632)-(634),  and  (635)  and  (636)  respec- 

TT 

tively.     In  the  first  table  we  have  given  1000  -zr~-  in 


order  to  avoid  the  small  numbers  which  result  from 
H 
SRn' 


Diameter  8  for  number  of  wires  /  = 

P 

S 

H 
Sv 

H 

JOOO  

±hn 

36 

42 

48 

60 

72 

O.O2O 

0.0184 

o  0172 

0.0156 

0.0140 

0.0054 

O.COOOIO 

0.000088 

0.024 

0.0220 

0.0208 

0.0184 

0.0168 

0.0078 

0.000015 

0.000123 

O.O28 

0.0260 

0.0244 

0.0216 

0.0!  96 

0.0107 

O  .  OOOO2O 

O.OOOI75 

0.032 

0.0296 

0.0276 

0.0248 

o  0228 

0.0139 

o  .  000026 

O.OOO229 

0.036 

0.0332 

0.0312 

0.0280 

0.0256 

0.0176 

0.000033 

0.000281 

o  040 

0.0368 

0.0348 

0.0308 

0.0284 

0  .  02  I  8 

0.000040 

0.000352 

0.048 

0.0444 

0.0416 

0.0372 

0.0340 

0.0313 

o  .  000060 

o  .  000492 

0.056 

0.0516 

0.0484 

0.0432 

0.0396 

0.0426 

0.000079 

o.ooo6S6" 

o  064 

0.0592 

0.0556 

0.0484 

0.0452 

0.0557 

0.000103 

0.001072 

0.072 

0.0664 

0.0624 

0.0556 

0.0508 

0.0705 

0.000131 

0.001125 

0.080 

o  .  0740 

0.0692 

0.0620 

0.0564 

0.0870 

0.000160 

0.001389 

o  088 

0.0812 

0.0764 

0.0680 

0.0624 

0.1331 

0.000195 

0.001688 

0.096 

0.0888 

0.0832 

0.0744 

0.0680 

o  .  i  408 

0.000232 

0.002004 

o.  104 

o  .  0960 

o  .  0900 

o  .  0804 

0.0736 

0.1471 

0.000272 

0.002356 

O.II2 

0.1036 

0.0968 

0.0868 

0.0792 

0.1586 

0.000306 

0.002725 

O.  12O 

0.1108 

0.1040 

0.0928 

0.08-18 

0.1958 

o  .  000364 

0.003329 

In  metallic  transmission-cables,  wires  of  less  than  O.O2 
inch  or  more  than  0.08  inch  diameter  are  very  seldom 


CALCULATION  OF  DIAMETERS  OF  CABLES.      2O3 


used.  The  values  of  d  given  in  these  two  tables,  in 
the  second  to  the  fifth  columns,  are  taken  from  values 
contained  in  the  first  column,  and  should  in  practice  be 
taken  in  round  numbers.  The  quality  of  the  metal 
used  for  transmission-cables  is  of  first  importance,  from 
the  fact  that  only  superior  qualities  can  withstand  for 
any  length  of  time  the  rapid  wear  to  which  the  cables 
are  subjected.  Swedish  iron,  which  possesses  at  the 
same  time  a  remarkable  tenacity  and  great  strength,  is 
especially  adapted  for  the  wires  of  transmission-cables. 
In  order  to  reduce  as  much  as  possible  the  number  of 
joints,  only  long  wires  should  be  used.  Experience  has 
shown  that  for  transmission-cables  wires  of  steel  offer 
no  advantages  over  those  of  good  iron. 


Diameter  of  wire  8  for  number  of  wires  i  — 

f(/v) 

s    H 
S     n 

36 

4* 

48 

60 

72 

0.020 

0.0188 

0.0180 

o.oi6& 

0.0160 

1554 

0.025 

O.O24 

0.0228 

0.0220 

0.0204 

0.0192 

2685 

0.043 

0.028 

O.O264 

0.0256 

0.0236 

0.0224 

4264 

0.068 

0.032 

o  .  0304 

O.O292 

0.0268 

0.0252 

6365 

O.IOI 

0.036 

o  0340 

0.0328 

o  .  0304 

0.0284 

9062 

0.144 

0.040 

0.0380 

0.0364 

0.0336 

0.0316 

12431 

0.197 

0.048 

0.0456 

0.0436 

0.0404 

0.0380 

21481 

0.341 

0.056 

0.0532 

0.0508 

0.0472 

0.0444 

34112 

0.542 

0.064 

o  .  0608 

0.0580 

0.0540 

o  0508 

50919 

0.894 

O.O72 

0.0684 

0.0656 

o  0608 

0.0572 

72499 

I.I52 

0.080 

0.0764 

0.0728 

0.0676 

0.0636 

99451 

1.580 

0.08S 

0.0836 

o  .  0800 

0.0744 

0.0700 

132369 

2.103 

0.096 

0.09.12 

0.0872 

0.0808 

0.0760 

171851 

2.730 

0.104 

0.0988 

0.0944 

0.0876 

0.0824 

218493 

3-471 

O.  112 

0.1064 

0.1016 

0.0944 

0.0888 

272892 

4-335 

O.  I2O 

0.1140 

0.1092 

O.IOI2 

0.0952 

335646 

5.332 

In  the  formulas  (632)-(634)  the  radius  R  of  the  pul- 
leys is  supposed  to  be  known  ;  the  values  of  d  given 


204  BELTS  AND  PULLEYS. 

r> 

by  them  are  admissible   only  when  the   ratio  -~-  gives 

for  the  tension  s  a  value  which,  added  to  S,  does  not 
exceed  25601.4  pounds.  In  the  case  where  j-f-^  ex- 
ceeds this  limit,  it  is  convenient  to  begin  the  calculation 
by  giving  to  7?  a  greater  value.  To  make  use  of  the 
preceding  formulas  and  tables,  we  must  begin  by  fixing 
upon  a  value  for  the  tension  *S.  This  may  easily  be 
done  with  the  aid  of  the  considerations  contained  in 
the  following  paragraph,  and  in  the  examples  which 
we  now  give  we  shall  suppose  this  preliminary  opera- 
tion already  accomplished. 

Example. — It  is  proposed  to  transmit,  by  means  of  a 
metallic  cable  running  over  pulleys  9.84  feet  in  diame- 
ter, a  force  of  550  pounds  :  required  the  proper  diameter 
for  the  wires  of  the  cable,  supposing  the  number  to  be 
i  =  36. 

If  we  take  5  =  9956.1,  we  shall  have  -=  =  - 

•j       9950-1 

0.0552,  which  in  the  first  table  (column  6,  line  9)  cor- 
responds to  a  diameter  of  $  —  0.064  inch.  From 

this  we  obtain  -^  =  — '-?—  =  022,  which,  in  the  table  on 
o       0.064 

page  201,  corresponds  nearly  to  S  =  9956.10,  and  is 
therefore  admissible.  If  we  had  taken  R  =  48  inches, 

r>  ,  o 

we  should  have  had  -F  =  — ^—  =  750 — a  value  less 
o        0.004 

than  the  limit  mentioned  above,  and  it  would  therefore 
be  necessary  to  increase  the  value  of  fi. 

Example. — The  force  transmitted  by  a  metallic  cable 
is  300  horse-power,  and  the  velocity  v  =  82  feet  per 


CALCULATION  OF  DIAMETERS  OF  CABLES. 


second;   taking  S  =    11378.4,  and  consequently  s  = 

TT 

25601.4  —  11378.4  =  14223,  we  shall  have  -^r-  = 

^TJ 

OQQ 

;-  =  0.000322.     In  the  first  table  the  near- 


11378.4  X  82 

TT 

est  value  of -~- is  0.000306  (column  7,  line  15).     The 

oZ' 

diameter  for  the  wires  is  therefore  d  =  0.112  inch  for 
i  =36,  d  =.  0.0868  inch  for  i  =  60.  For  the  value 
s  —  14223,  we  have,  for  the  radius  of  the  pulleys,  R  = 

14223000  X  0.0848 

=  84.8  inches.     1  he  expression  v  = 

14223 

2rtRn 
rr-f-  gives  for  the  number  of  revolutions  per  minute 

1 2  X  OO 

82  X  12  X  60 

:-6^T4:8-  =UI- 

Example. — It  is  required  to  calculate  the  horse-power 
which  may  be  transmitted  by  a  cable  of  thirty-six  wires, 
the  diameter  of  the  wires  being  0.08  inch,  the  diame- 
ter of  the  pulleys  9.84  feet,  and  the  number  of  revolu- 

R       59.04 

tions  per  minute  go.     In  this  case  we  have  -$  •=.        _ 

tf        0.08 

T  A  *)  *}  'JOOO 

=  73^,  which,  from  formula  (637),  gives  s  =  — - — ^~ 
•  =  19272.3  and  5  —  6329.1.     For  d  =  0.08  and  i  =  36, 

TT 

the     first     table    furnishes     the    value     1000-^75-  = 

SRn 

0.001389;  hence 

H_  0.001389  _  0.001389  X  6329.1  X  59-°4  X  9Q 

1000  1000 

=  46.71  horse-power.    With  a  pulley  of  8  feet  diameter 


206  BELTS  AND   PULLEYS. 

R       48  14223000 

we  would  have  -?  =  — TT  =  ooo,  s  = 7 =  23705, 

o       0.08  600 

S  =  1896.4.     Consequently 

0.001389  X  1896.4  X  4B  X  90 
H  =  —  —=  1 1. 40  horse-power. 

1000 

Example. — Upon  the  driven  arbor  of  a  transmission 
by  cable  a  resistance  of  no  pounds  acts  continuously 
with  a  lever  arm  of  40  inches.  Required  the  proper 
diameter  for  the  36  wires  of  the  cable,  supposing  we 
give  to  the  pulleys  the  smallest  admissible  radius.  In 
order  to  satisfy  this  last  condition,  we  ought  to  take 
(from  what  precedes)  s  —  17067.60  and  5  =  8533.80, 

which  gives  -^  (PR)  =  2  X  1 10  X  40  =  8800.     In  the 
•^ 

second  table  (column  6,  line  5)  we  find,  for  the  nearest 

value  of --(PR),  d  =  0.036  inch.     From   the  table  on 

•p 
page  201,  therefore,  we  obtain  -~  =  833,  R  =  833  X 

0.036  =  30  inches. 

Example. — A  cable  of  42  wires  transmits  a  force  of 
30  horse-power  at  a  velocity  of  100  revolutions  per 
minute.  Required  the  proper  diameter  for  the  wires 
of  the  cable,  taking  5  =  8533.80.  In  this  case  s  — 

,  s  H       17067.60       30 

17067.60,  and  -~  —  =  -  -  X  -  -   —  0.6.    The  sec- 

5  n        8533.80        100 

ond  table  gives,  for  the  nearest  value  of  -~  —  to  0.6, 

o    fl 

d  —  0.056  inch.     From  formula  (637),  then,  we  have 
for  the  radius  of  the  pulleys  R  =  0.056  X 
833  X  0.056  =  46.65  inches. 


DEFLECTIONS  IN  A    CABLE.  2O/ 


§21.     Deflections  in  the  Cable  of  a  Horizontal  Transmission. 

In  order  that,  in  the  two  parts  of  a  transmission- 
cable,  the  tensions  T  and/  have  proper  values  (not  too 
small,  for  then  the  cable  will  slip  on  its  pulleys;  nor 
too  great,  because  the  wear  is  then  great),  the  deflec- 
tion which  we  give  to  each  part,  in  a  state  of  repose, 
must  be  a  determined  quantity.  It  is  equally  necessary 
that  we  know  the  deflections  which  are  produced  dur- 
ing the  motion  of  the  cable,  in  order  to  leave  sufficient 
room  for  the  passage  of  the  cable.  The  deflection  of 
a  cable  depends  upon  the  tension  of  its  wires. 

Let  us  represent  by 

A  the  separation  of  the  pulleys  of  a  horizontal  trans- 
mission in  feet;  h  the  deflection  of  the  cable  in  feet 
(hl  for  the  driving  part,  /z2  for  the  driven  part,  and  7/0 
for  the  state  of  repose)  ;  S  the  tension  per  square  inch 
in  the  wires  (Sl  for  the  driving  part,  52  for  the  driven 
part,  and  S0  for  the  state  of  repose). 

For  a  metallic  cable  of  any  number  of  wires  we  have 
the  relations 


•3  =  0-3535  [o.369§  -  y  (0-369  §J*  -  i     (638) 


and 


1  =  3-8029^+^).  .  .  .  (639) 

By  means  of  these  formulas  the  following  table  has 


2O8  BELTS  AND  PULLEYS. 

been  calculated.     As  a  first   approximation  we    may 
take  simply 

h  A 

- =  04755    --.    ....    (640) 


In    order  to   make  use  of   the  table,  we  begin   by 

A 

determining  from  the  given  quantities  the  ratio  -~-  of 

the  separation  of  the  pulleys  to  the  tension  developed 
in  the  wires,  and  then  find  in  the  table  the  number 
nearest  to  this  ratio.  From  this  we  obtain  the  value 

of  —j ,  which  gives  the  amount  of  deflection  h.     The 
^i 

tension  50  of  the  cable  in  a  state  of  repose  is  not  the 
arithmetical  mean  between  5^  and  S^ ;  we  may,  by  a 
more  complicated  calculation,  however,  determine  it 
from  the  length  of  the  two  cable  parts.  The  value 
which  we  need  to  know  is  the  deflection  h0  in  the  two 
parts  of  the  cable  for  a  state  of  repose,  and  we  have 
approximately 


0.287*,.    .    (641) 


This  expression  gives  for  hQ  a  value  slightly  too 
great,  but  which  approaches  more  nearly  the  true  value 
as  the  tensions  5,  and  52  become  less.  The  error  may 
be  still  farther  decreased  by  using,  instead  of  exact 
values  of  A,  and  h»  those  furnished  by  formula  (640). 

The  driving  part  of  the  cable  does  not  necessarily 


DEFLECTIONS  IN  A    CABLE. 


209 


occupy  the  higher  position,  as  is  the  case  in  Fig.  77 : 
it  may  be  placed  in  the  lower  position,  as  in  Fig.  78. 
In  the  latter,  the  space  required  by  the  deflection  of 


FIG.  77. 

the  cable  is  considerably  less  than  in  the  former.  The 
two  parts  of  the  cable  do  not  intersect  each  other  as 
long  as  Az  —  h^  <  2R.  With  a  cable  in  motion,  we 
may  place,  at  the  lowest  point  of  the  curve,  a  gradu- 
ated rule,  by  means  of  which  we  may  observe  at  any 
instant  the  tensions.  The  graduation  of  the  rule  may, 
moreover,  be  such  as  to  give  directly  the  tension  5. 
14 


210 


BELTS  AND  PULLEYS. 


TABLE  OF  DEFLECTIONS  IN  METALLIC  CABLES. 


h 

A 

A 
S 

h 

A 

A 
s 

h 
A 

A 
S 

h 

~A 

A 
S 

0.003 

0.006 

0.033 

0.069 

0.063 

O.I2S 

0.093 

0.183 

0.004 

0.008 

0.034 

0.071 

0.064 

o.  130 

0.094 

0.185 

0.005 

O.OII 

0.035 

0.073 

9-065 

0.132 

0.095 

0.186 

0.006 

0.013 

0.036 

0.075 

0.066 

0.134 

0.096 

0.188 

0.007 

0.015 

0.037 

0.077 

0.067 

0.136 

0.097 

o.  190 

0.008 

0.017 

0.038 

0.079 

0.068 

0.138 

0.098 

0.191 

O.OOg 

0.019 

0.039 

0.081 

0.069 

0.140 

0.099 

0.193 

0.010 

0.021 

0.040 

0.083 

to  .  070 

0.142 

0.100 

0.195 

O.OII 

O.O23 

0.041 

0.085 

0.071 

0.144 

O.IOI 

0.196 

•  0.012 

0.025 

0.042 

0.087 

0.072 

0.145 

O.IO2 

0.198 

0.013 

O.027 

0.043 

0.089 

0.073 

0.147  ! 

0.105 

0.203 

0.014 

0.029 

0.044 

0.091 

0.074 

0.149 

O.IIO 

0.2II 

0.015 

O.03I 

0.045 

0.093 

o  075 

0.151 

0.115 

0.219 

0.016 

0.034 

0.046 

0.095 

o  076 

0.153 

0.120 

0.226 

0.017 

0.036 

0.047 

0.097 

0.077 

0    T55    ! 

0.125 

0.234 

0.018 

0.038 

0.048 

0.099 

0.078 

0.156 

O.I3O 

0.24T 

0.019 

0.040 

0.049 

O.  IOI 

0.079 

0.158 

0.135 

0.248 

0.020 

0.042 

0.050 

0.103 

0.080 

0.160 

O.T40 

0-255 

0.021 

0.044 

0.051 

o.  105 

0.081 

0.162 

0.145 

0.261 

0.022 

0.046 

0.052 

o.  107 

0.082 

0.164 

0.150 

0.267 

0.023 

0.048 

0-053 

0.109 

0.083 

0.165 

0.155 

0.274 

0.024 

0.050 

0.054 

0.  Ill 

0.084 

0.167 

0.160 

0.279 

O.025 

O.052 

0-055 

0.113 

0.085 

0.169 

0.165 

0.285 

0.026 

0.054 

0.056 

o.  115 

0.086 

0.171 

o.  170 

0.291 

0.027 

0.056 

0.057 

0.117 

0.087 

0.173 

o.i75 

0.296 

0.028 

0.059 

0.058 

0.119 

0.088 

0.174 

0.180 

0.301 

O.O29 

0.061 

0.059 

O.I2I 

0.089 

0.176 

0.185 

0.305 

0.030 

0.063 

O.o6o 

0.123 

0.090 

0.178 

o.  190 

0.310 

0.031 

0.065 

0.061 

0.125 

0.091 

0.179 

0.195 

0.315 

0.032 

0.067 

0.062 

0.127 

0.092 

0.181 

0.200 

0.319 

Example. — In  the  last  example  of  §  20  the  separation 
A  of  the  pulleys  is  360.8  feet,  and  we  take  the  tension 
S\  —  8533.8  pounds  per  square  inch.  Required  the 
deflections  in  the  parts  of  the  cable.  For  the  driving 

A         360.8 
part  of   the  cable   the  relation    ~  =  ^777-5  —  0.0423 


DEFLECTIONS  IN  A    CABLE.  211 

corresponds  in  the  table  (column  2,  line  18)  to  the  value 
—  —  0.02.  Hence  we  have  Al  =  360.8  X  0.02  —  7.216 
feet.  For  the  driven  part  of  the  cable  we  have  from 

O   £   .3   .3      O 

formula  (631)  5  = =  4266.9,  and  consequently 

2 

A         260  8  A 

-~  =  — -^7—  =  0.0845.  For  this  value  of  -~  the  table 
5  4266.9  5 

gives  (column  4,  line  9)  -j-  =  0.041,  and  we  have  7z2  = 

A 

360.8  X  0.041  =  14.79  feet-  From  formula  (641)  the 
deflection  of  the  cable  in  a  state  of  repose  is  AQ  = 


FIG.  78. 

0.67  X  14.79  +  °-28  X  7.216  =  11.93  feet.  We  have 
also  A.2  —  Al  —  14.79  "  7-2i6  =  7-574  and  2R  = 
2  X  3.8875  =  77750  feet.  Since  2R  >  h^  —  A19  we  may 
if  necessary  make  use  of  the  disposition  of  Fig.  78. 
(See  first  example  of  §  22.) 

Example. — In  the  third  example  of  §  20  the  distance 
bfelow  the  line  of  centres  of  the  pulleys  is  9.84  feet ;  it 
is  required  to  determine  the  proper  distance  between 
the  pulley-centres.  Assuming  that  we  can  make  use 
of  the  disposition  represented  in  Fig.  78,  the  greatest 
admissible  value  for  the  distance  of  separation  of  the 
pulleys  may  be  calculated  from  the  deflection  of  the 
cable  while  in  a  state  of  repose.  Making  use  of  the 
approximate  formula  (640),  and  remembering  the  value 


212  BELTS  AND  PULLEYS. 

5,  —  6642.141  pounds  per  square  inch,  we  shall  have 

A* 

—  and   h,  =  2//,.      Formula  (641) 


then  gives 

_  (0.67  X  2  +  0.28)^"  X  0.4755 

A.  =   9.84  -  6647.747" 


/       0.84 
=  V  5^7S5^ 


x6642.141 


..  ,  --  - 

=  =  i/84847'23  = 

feet. 

§  22.     Transmission  by  Cable  with  Increased  Tension. 

When  the  pulleys  of  transmission  are  very  distant 
from  each  other  the  deflections  given  by  the  preced- 
ing formulas  become  so  great  that  it  is  often  necessary 
to  place  the  pulleys  at  a  great  elevation,  or  to  provide 
a  deep  trench  for  the  cable  when  we  wish  to  dispense 
with  intermediate  pulleys  and  guides  (see  §  28).  In  a 
great  many  cases  we  may  arrive  at  the  same  result  by 
simply  giving  to  the  cable  a  greater  tension  than  is 
necessary  to  prevent  slipping,  and  taking  care  to  give 
to  the  cable  a  diameter  sufficient  to  withstand  the  ad- 
ditional strain.  This  artifice  may  be  employed  all  the 
more  easily  when  the  transmission  is  to  be  used  for 
moderate  forces,  and  consequently  a  small  diameter  of 
the  cable  is  sufficient.  It  is  only  necessary  to  examine 
carefully  the  rules  which  follow,  to  be  convinced  that 
a  rational  use  of  this  method  presents  in  reality  little 
or  no  difficulty. 

A  transmission  by  cable,  established  under  the 
above  conditions,  constitutes,  by  way  of  distinction 
from  ordinary  cable-transmission,  what  we  term  a 


TRANSMISSION    WITH  INCREASED    TENSION.   213 

transmission  with  increased  tension.  We  may  distin- 
guish it  from  ordinary  transmissions  by  giving  the  sign 
s  to  the  forces  and  dimensions  connected  with  it 
(T8,  4,  Ss,  3S  instead  of  T,  t,  5,  and  6).  The  tension 
T,  in  the  ordinary  mode  of  transmission,  ought  not  to 
be  less  than  2P;  in  a  transmission  with  increased 
tension  the  tension  ought  to  be  increased  by  a  certain 
factor  which  we  shall  designate  by  m.  We  shall  there- 
fore have 

Ta  =  mT,         tB=(2in-i)t,        A  =  -2i—  I.     (642) 

The  tension  5,  in  the  driving  part  of  the  cable  is  not 
changed,  but  in  the  driven  part  the  tension  Szs  is  no 

longer  equal  to     '.     We  take  instead 


The  diameter  38  of  the  wire  is  deduced  from  the 
diameter  3  given  by  one  of  the  formulas  (632)  to  (634), 
by  means  of  the  relation 


3S  =  3  Vm.      .....     (644) 

If,  however,  3  is  calculated  from  formula  (636)  or  (638), 
we  must  take 


38  =  3tym.      .....     (645) 

From  these  formulas  the  following  table  has  been 
calculated.  It  is  important  to  remark,  that  in  cables 
with  increased  tension  the  strain  in  the  wires  is  no 


214 


BELl^S  AND  PULLEYS. 


greater  than  in  ordinary  cables,  because  they  have  a 
proportionately  greater  diameter.  The  cable  is  heavier 
in  the  former  than  in  the  latter  case,  and  should  there- 
fore be  strained  more  firmly  over  the  pulleys  in  order 
to  reduce  the  deflection  in  the  driven  part. 


T8 
m=T 

7s 
P 

t»          t8        S28 

t  ~  P  ~  S2 

S2S_  ts 
Si  ~  7  a 

SS           A<- 

y  =  \m 

y  -  \fm 

1.2 

2.4 

1.4 

0.58 

I.  10 

.06 

1.4 

2.8 

1.8 

0.64 

1.18 

.12 

1.6 

3-2 

2.2 

0.69 

1.26 

•17 

1.8 

3.6 

2.6 

0.72 

1-34 

.22 

2.0 

4.0 

3-o 

0-75 

1.41 

.26 

2.2 

4.4 

3-4 

0.77 

1.48 

•3° 

2.4 

4.8 

3-8 

0.79 

•55 

•34 

2.6 

5-2 

4.2 

0.81 

.61 

•38 

2.8 

5.6 

4.6 

0.82 

.67 

.41 

3-o 

6.0 

5-0 

0.83 

•73 

1.44 

3-2 

6.4 

5-4 

0.84 

•79 

1.47 

3-4 

6.8 

5.8 

0.85 

.84 

1-50 

3-6 

7.2 

6.2 

0.86 

.90 

1-53 

3-8 

7.6 

6.6 

0.87 

•95 

1.56 

4.0 

8.0 

7.0 

0.88 

2.00 

i-59 

4.2 

8.4 

7-4 

0.88 

2.05 

1.61 

4.4 

8.8 

7.8 

0.89 

2.  IO 

.64 

4.6 

9.2 

8.2 

0.89 

2.14 

.66 

4.8 

9.6 

8.6 

0.90 

2.19 

.69 

5-0 

10.  0 

9.0 

0.90 

2.24 

•71 

5-5 

II.  0 

10.  0 

0.91 

2.36 

•75 

6.0 

12.0 

II.  0 

0.92 

2-45 

.82 

6.5 

13.0 

12,0 

0.92 

2.55 

.87 

7  o 

14.0 

13-0 

o-93 

2.65 

.91 

7-5 

15-0 

14.0 

0-93 

2.74 

.96 

8.0 

l6.O 

15-0 

0.94 

2.83 

2.00 

Example. — In  the  first  example  of  §21  the  driven 
part  of  the  cable  has  a  deflection  of  /z2  =  11.76  feet, 
and  the  diameter  of  the  wire  is  0.056  inch.  If  we  wish 
to  diminish  the  value  of  //2  by  using  a  cable  with 
increased  tension,  the  value  of  8  must  be  increased  ac- 
cordingly. If  we  take  m  =  2,  the  table  gives  (col- 


TRANSMISSION   WITH  INCREASED    TENSION.    21$ 

umn  4,  line  5)  -f  =  0.75,  S2S  =  0.75  X  8533.8  =  6400.35 


A 

pounds.  Consequently  -~-  =  ^  -  -  —  =  0.056,  which, 

in  the  table  of  §21,  corresponds  to  —  T-   =   0.027   or 

^i 

h  =  0.027  X  360.8  =  9.74  feet.  The  tension  of  de- 
flection s  has  the  same  value  as  if  for  an  ordinary  ca- 

ble ;  the  quotient  -~r--  does  not  change  its  value,  and 
o  i  i  & 

consequently  d  may  be  determined  by  means  of  formu- 
la (636).  The  preceding  table  gives,  then,  ds  =  1.26^ 
=  1.26  X  0.56  =  0.07  inch. 

When,  in  calculating  the  diameter  $  for  an  ordinary 
cable  of  36  wires,  we  obtain  a  very  small  value,  the 
cable  itself  may  have  such  a  small  diameter  that  its 
manufacture  involves  as  great  an  expense  as  for  a  cable 
of  larger  diameter.  In  such  a  case  we  cannot  recom- 
mend too  highly  the  use  of  a  transmission  by  cable 
with  increased  tension,  which  has  the  advantage  of  re- 
ducing the  deflection  in  the  driven  part  of  the  cable 
without  appreciably  increasing  the  expense  of  manu- 
facture. As  a  general  rule,  we  should  never  make  use 
of  wires  of  a  less  diameter  than  0.04  inch,  so  that  the 
minimum  diameter  of  cable  may  be  0.32  inch. 

Example.  —  For  a  transmission  by  cable,  we  have 
given  H  =  5.5,  n  =  100,  and  A  —  590.4.  If  we  as- 

sume  vS,    =    14223    and  s  —    11378.4,  we  have  -~-  - 

Oj     11 

=  —  2i-3  x  J>15-  —  0.044,  which,  for  i  =  36  (table  on 
14223        ioo 

page  203)  gives,  for  the  diameter  of  the  wire  $  —  0.024 


2l6  BELTS  AND  PULLEYS. 

\\i     t,  A         59°-4  A 

inch.     We  have  also  -          —       =  0.0415,    --   = 


'  -  —  0.0830,  and  consequently,  from  the  table  of 

page  210,  /i,  =  0.0198    X    590.4  =    11.69  feet,  Aa   = 
0.04  X  590.4  =  23.616  feet,  A9  —  h,  —  23.616  —  11.69 


=  11.926  feet.      But  since  R  =  -  d    =    1250 

113704 

X  0.024  —  30  inches,  A9  —  A,  is  greater  than  2R.  In 
this  case,  therefore,  we  cannot  place  the  driven  part  of 
the  cable  above  the  driving  part,  and  the  axes  of  the 
pulleys  must  have  a  height  above  the  ground  at  least 
equal  to  R  +  h*  =  2.5  +  23.62  =  26.12  feet.  Sup- 
pose  now  we  take  for  the  cable  diameter  0.32  inch, 
instead  of  8  X  0.024  =  0.192  inch;  that  is,  we  take 
0.04  inch  for  the  diameter  of  the  wires.  We  have  then 

<5S        0.04 

-~-  —  -  =  1.67,  and  the  preceding  table  gives  (col- 

umns 6  and  4,  line  18)  52S  =  14223  X  0.89  =  13058.47. 

Consequently  ~^-  =  -  -  —  0.0452  and  A98  =  0.0228 

->2S      13055.47 

X  590.4  =  13.46  feet,  A,,  —  A,  =  13.46  —  11.69  =  1.77 
feet.  As  before,  R  =  1250  X  <$s  =  S°  inches  and  2R 
-•=  8.33  feet:  the  inequality  h^  —  A,  <  2R  is  now  sat- 
isfied, and  we  may  give  to  the  cable  the  desired  ar- 
rangement. The  maximum  deflection  in  this  case  cor- 
responds to  the  state  of  repose,  for  which  we  have,  from 
formula  (641),  Aos  =  12.28  feet.  The  height  of  the  pul- 
ley-axes above  the  ground  must  be  at  least  A08  +  R 
—  12.28  +  4.165  =  16.445  feet;  that  is,  less  by  nearly 
10  feet  than  for  the  first  calculated  cable. 


TRANSMISSION  BY  INCLINED   CABLE. 


217 


§  23.     Transmission  by  Inclined  Cable. 

Of  the  various  transmissions  by  metallic  cable,  the 
one  which  has  met  with  the  greatest  development  cor- 
responds to  the  case  in  which  the  pulleys  are  not  on 
the  same  level,  one  being  higher  than  the  other,  and 
constitutes,  therefore,  what  we  call  an  inclined  trans- 
mission. We  give  here  the  rules  necessary  for  such 
transmissions.  In  the  cable  B  CD,  Fig.  79,  which  rep- 


ni 


a' 


resents  a  part  of  an  inclined  transmission,  the  summit 
of  the  curved  axis  is  not  in  the  middle  of  the  distance 
between  the  points  of  suspension,  and  the  deflections 
are  therefore  different  from  those  in  the  cable  of  a 
horizontal  transmission.  The  deflections  may,  how- 
ever, be  easily  determined  in  functions  of  the  elements 
of  a  horizontal  transmission,  having  the  same  separa- 
tion of  pulleys  and  sensibly  the  same  tensions. 

Let  us  represent  by 

h  and  A,  respectively,  the  deflection  of  the  cable 
and  the  separation  of  the  pulleys  of  a  horizontal  trans- 
mission ; 


2l8  BELTS  AND  PULLEYS. 

S  the  tension  corresponding  to  the  point  of  sus- 
pension of  the  part  of  the  cable  under  consideration  ; 

//  and  ti  ',  respectively,  the  smallest  and  greatest  de- 
flection (FC  and  EC)  in  an  inclined  transmission,  in 
which  the  separation  of  the  pulleys  measured  horizon- 
tally is  equal  to  A  ; 

a'  and  a"  ,  respectively,  the  distances  CB^  and  CDl  of 
the  summit  of  the  curve  from  verticals  through  the 
points  of  suspension  ; 

S/  and  S",  the  tensions  (at  B  and  D)  at  the  lower 
and  higher  points  of  suspension  respectively  ; 

H  the  difference  between  the  levels  (EF)  of  the 
points  of  suspension. 

The  values  of  //  and  S  may  be  determined  by  means 
of  the  rules  already  given.  We  have  then 

H-  ,        A"  =  H  +  K  ;      (646) 


//7  -  A), 
5/7    -  S'  -  3.804^.      •     •     (648) 

In  certain  cases  the  value  of  a'  maybe  negative  ;  the 
summit  of  the  curve  of  the  cable  prolonged  is  then  sit- 
uated beyond  the  lower  pulley.  The  tension  of  flec- 
tion s,  and  consequently  the  diameter  of  the  pulleys, 
are  determined  when  we  have  obtained  the  value  of  the 
tension  S",  which  very  often  does  not  differ  materially 
from  5.  The  difference  between  the  two  tensions  be- 


TRANSMISSION  BY  INCLINED   CABLE.  2  19 

comes  important  only  in  cases  where  several  inclined 
transmissions  are  taken  from  a  single  higher  pulley. 

Example.  —  A  transmission  by  cable,  the  data  of 
which  are  the  same  as  in  the  fifth  example  of  §  20,  has 
its  pulleys  placed  at  different  heights  ;  taking  for  the 
difference  in  the  levels  of  the  pulleys  H  =  16.4  feet,  it 
is  required  to  determine  the  deflections  and  the  posi- 
tions of  the  curve-summits. 

For  the  driving  part  of  the  cable  we  have 

5*1  =  8533.8,       7*i  =  7.216  feet,       H  —  16.4  feet,       A  —  360.8  feet. 

Starting   at  the  lower  pulley,  we  have,  from  formula 


16.4 

--  1.35  feet, 


2 

//",  —  7-2i6  -f-  1.35  =  8.566  feet ; 

S6o.8/          I     16.4  \ 
a\  --     *—\l  -  -   -  ^jgj  =  1 80.4X0  432  =  77-93  feet, 

a'\  ==  360.8  +  77.93  =±  382.87  feet. 
For  the  driven  part  of  the  cable, 

S,  =  4266.9,     h,  =  14.79  feet ; 
consequently 

/          i     16.4"  \        16.4 

k\  ==  I4-79V1  +  16  T^TO?;  "    ~2~    =  7*73  feet> 
h'\  =  16.4  +  773  =  24.13  feet. 


22O  BELTS  AND  PULLEYS. 

For  the  state  of  repose, 

hQ  =  0.67  X  14.79  +  °-28  X  7.216  =  12.05  feet  ; 
hence 


.  16.4 

'"'+-    ~-    ^  5-24  feet, 


a\  = 


tif.  —  16.4  +  5.24  —  21.64 

360.8  /         i    i6.4\ 
-  ---   i  —  ^j  =  119.06  feet, 

4  I2.05/ 


2     \        4  I2.05 
a'\  —  360.8  --  119.06  =  241.74  feet. 

The  tensions  in  the  driving  part  of  the  cable  are  as 
follows:  S\  =  8533.8  -  (7.216-  1.35)3.804  =8511.5, 
S",  =  8533-8  +  (8.566  -  7.216)3.804  =  8538.94;  the 
values  of  S\  and  S'\  differing  so  slightly  from  Sl  that 
we  may  neglect  the  difference. 

The  heights  which  the  calculations  furnish  for  the 
deflections  of  an  inclined  transmission  should  be  laid  out 
in  the  drawing  to  a  scale  three  or  five  times  that  of  the 
horizontal  lines  ;  we  then  trace  the  curve  of  the  cable 
as  an  arc  of  a  parabola  (see  the  following  paragraph), 
and  try  if  the  conditions  of  the  ground  will  permit  us 
to  use  the  curve  obtained.  If  this  prove  not  the  case, 
we  must  recommence  the  calculation  by  adopting  new 
values  for  the  tension  until  we  have  obtained  a  curve 
which  will  satisfy  the  conditions.  With  a  little  prac- 
tice, it  is  easy  to  determine  by  the  eye  the  proper  val- 
ues to  be  adopted,  and  the  calculation  may  then  be 
made  without  difficulty. 


METHOD  OF  TRACING  THE  CURVES  OF  CABLES.    221 


§  24.     Method  of  Tracing  the  Curves  of  Cables. 

The  curve  of  a  cable  may  be  drawn  with  sufficient 
accuracy  for  ordinary  purposes  by  assuming  it  to  be  an 
arc  of  a  parabola.  After  having  determined  the  summit 
C  of  the  part  of  the  cable  BCD,  Fig.  80,  as  explained 
in  the  preceding  section,  divide  into  two  equal  parts, 
at  the  points  C,  and  C2,  the  two  distances  B^C  and 
D^C l  (BlDl  being  tangent  to  the  curve  of  its  summit), 
and  through  the  points  C^  and  C2  draw  the  lines  BCl 
and  DC.,,  which  give  the  directions  in  which  the  cable 


leaves  the  pulleys.  Divide  the  distances  CCV  and 
into  a  certain  number  of  small  equal  parts  at  the 
points  i,  2,  3,  etc.,  and  I,  II,  III,  etc.;  by  joining  il, 
.2!!,  3!!!,  etc.,  we  obtain  a  series  of  lines  tangent  to 
the  required  parabola.  By  a  similar  method  with  CC^D 
we  obtain  the  other  part  of  the  curve.  When  the  sum- 
mit C  of  the  curve  falls  outside  of  the  pulleys,  on  the 
side  of  the  pulley  which  occupies  the  lower  level,  apart 
of  the  parabola  near  the  summit  cannot  be  made  use 
of.  but  the  construction  is  still  the  same. 


222  BELTS  AND  PULLEYS. 


§  25.     Transmission  by  Cable  with  Pulleys  near  together. 

When  the  distance  between  the  pulleys  of  a  trans- 
mission by  cable  is  small,  it  is  necessary,  first  of  all, 
that  the  deflections  have  not  too  small  values,  in  order 
that  the  cable  may  run  properly  upon  the  pulleys,  and 
also  that  we  may  be  able  to  shorten  the  cable  without 
seriously  increasing  the  tension.  We  adopt  then  for 
S1  a  very  small  value,  and  thus  determine  upon  a  value 
for  the  deflection  ;  then,  by  means  of  formula  (638) 
and  the  table  calculated  from  it,  obtain  S1  ;  /  and  R 
are  then  calculated  as  we  have  already  indicated.  For 
a  small  tangential  resistance  and  a  small  separation  of 
the  pulleys,  transmissions  by  cable  may  still  be  used 
with  satisfactory  results. 

Example. — A  metallic  cable  transmits  a  force  of  6 
horse-power  at  150  revolutions  per  minute  ;  the  separa- 
tion of  the  pulleys  is  65.6  feet  and  the  deflection  in  the 

driven  part  of  the  cable  3.28  feet.     We  have  then  — r 

A 

A 

—  0.05,  which,  from  §  21,  corresponds  to  -~  —  0.103, 

65.6 

and  we  obtain  S.  =  -      ~  =  637.     In  order  to  find  the 
0.103 

value  of  tf,  we  must  know  that  of  s  Assuming  that 
s  -|-  5*!  is  still  equal  to  25601.4,  we  have  s  =  25601.4 

s  H       24964.4    6 

—  637  —  24064.4,  which  gives  > =  -  -  = 

S,   n  637     150 

1.57.  The  second  table  of  §20  gives  (column  7,  line 
li),  therefore,  8  —  0.08  inch,  for  i  —  36.  From  for- 
mula (637)  we  have  for  the  radius  of  the  pulleys  R  = 


RIM  OF  CABLE-PULLEYS.  22$ 

I4223OOO 
0.08 ^ =  45.0  inches,     rrom  what  precedes,  we 

find  that  these  values  of  8  and  R  are  perfectly  admis- 
sible. If  we  wish  to  take  for  the  diameter  of  the  ca- 
ble, d  =  8#  =  0.48  inch,  that  is,  &  is  reduced  to  0.06 
inch,  it  is  only  necessary  to  give  to  R  a  smaller  value. 
In  this  case  the  table  of  §20  gives  (column  7,  lines  8 

and  9)  -~ =  0.718,  hence  s  =  0.7 iSS^  —  0.718  X 

o  j    r  I  £~1 

637-^-  =   11434.15,    and    formula  (637)    gives    R  = 

14223000 

o.oo- --  =  74  inches.     In  some  cases  pullevs  of 

11434.15 

large  radii  cannot  be  conveniently  used,  and  we  are 
obliged  to  use  pulleys  of  different  radii  in  order  to 
make  the  deflections  great  enough.  For  the  transmis- 
sion of  considerable  forces,  we  obtain  good  results  only 
on  the  condition  of  giving  to  the  pulleys  a  certain  ve- 
locity of  rotation,  the  limits  for  which  are  indicated  at 
the  end  of  the  following  paragraph. 


§  26.     Rim  of  Cable-pulleys. 

When  first  used,  the  rims  of  cable-pulleys  were  made 
,of  wood  covered  with  leather,  but  practice  soon  de- 
monstrated the  fact  that  rims  of  metal  are  preferable, 
and  at  the  present  time  the  latter  are  used  almost  ex- 
clusively in  all  cases  where  durability  forms  an  impor- 
tant factor.  Figs.  81  and  82  represent  two  cast-iron 
rims,  single  and  double.  The  sides  of  the  groove  in 
the  single  rim  are  inclined  at  an  angle  of  30°  with  the 
middle  plane  of  the  pulley.  In  the  double  rim  such 


224 


BELTS  AND  PULLEYS. 


an  inclination  would  necessitate  too  great  a  weight 
for  the  projection  between  the  two  grooves ;  the  in- 
clination of  the  sides  of  this  projection  is  therefore  less 
than  30°.  In  Fig.  82  (which  represents  a  portion  of  a 
large  pulley)  this  inclination  is  15°.  All  the  dimensions 
indicated  in  the  figures  are  in  terms  of  the  diameter  d 


FIG.  82. 


of  the  cable.  Since  cables  of  less  than  0.4  inch  diameter 
are  seldom  used,  we  may  consider  the  value  of  d  =  0.4 
inch  as  the  inferior  limit  of  the  unit  for  the  construction 
of  cable-pulleys.  The  grooves  in  the  faces  of  the  pul- 
leys are  bottomed  with  gutta-percha  driven  into  the 
dovetails,  as  shown  in  the  figures  ;  or  small  pieces  of 
wood,  which  are  introduced  into  the  dovetails  through 
openings  in  the  side  of  the  rim.  Fig.  82  shows  two 


RIM  OF  CABLE-PULLEYS.  22$ 

openings  of  this  kind  covered  up  by  pieces  which  are 
bolted  in  after  the  insertion  of  the  wooden  pieces.  Of 
late  years  grooves  with  leather  bottoms  have  come  in- 
to use  for  very  heavy  cables ;  to  this  end  old  belts  cut 
into  strips  and  wedged  into  the  dovetails  may  be  ad- 
vantageously used.  Professor  Fink  has  successfully 
employed  bottoms  formed  by  winding  twine  tightly 
around  in  the  dovetails ;  bottoms  thus  made  give  great 
resistance  to  slipping.  Bottoms  of  cork  have  also  been 
used,  but  while  they  offer  the  advantage  of  being  in- 
expensive, they  have  not  been  tested  sufficiently  in  prac- 
tice to  determine  their  utility  for  transmission  by  cables 
where  there  is  danger  of  slipping.  When  we  wish  to 
make  use  of  bottoms  of  twine,  the  depth  of  the  dove- 
tails need  not  be  so  great  as  that  indicated  in  the  fig- 
ures. In  the  first  three  modes  of  furnishing  the  grooves 
with  bottoms  which  present  more  resistance  to  slip- 
ping than  cast-iron  (gutta-percha,  wood,  and  leather), 
the  profile  of  the  groove  upon  which  the  cable  rests 
may  be  hollowed  out  after  the  introduction  of  the  ma- 
terial into  the  dovetails.  Pulleys  of  12  to  15  feet  in 
diameter  are  ordinarily  cast  in  two  pieces,  which  makes 
them  easier  to  handle  and  transport ;  projections  are 
cast  upon  the  inside  of  the  rim  by  means  of  which  the 
two  parts  may  be  bolted  together. 

In  order  that  no  harm  may  come  to  the  rim  through 
excessive  centrifugal  force,  the  velocity  of  rotation  of 
the  rim  should  not  exceed  100  feet  per  second.  The 
velocity  of  about  90  feet  per  second,  which  is  now  com- 
monly given  to  metallic  cables,  may  be  considered  as 
without  disadvantages  in  ordinary  practic^ 
15 


226  BELTS  AND  PULLEYS. 


§  27.     Arms  and  Nave  of  Cable-pulleys. 

The  body  and  rim  of  a  cable-pulley  are  ordinarily  of 
cast-iron,  as  is  often  the  case  with  the  entire  pulley. 
We  however  sometimes  find  arms  of  wrought-iron  set 
into  cast-iron  rims  (see  Fig.  96).  In  any  case  the  num- 
ber of  arms  A  is  determined  from  the  expression 


The  cross-sections  of  cast-iron  arms  are  oval  or  flanged  ; 
in  either  case  the  width  in  the  plane  of  the  pulley  is 
given  by  the  formula 

T       /? 

h  =  ^d  +  •    -j-  .....     (650) 
4  -^ 

In  a  flanged  cross  section  the  thickness  of  the  prin- 
cipal flange  (in  the  plane  of  the  pulley)  is  e  =  —  ,  and 

that  of  the  secondary  flange  e'  =  \e.  In  an  oval  cross- 
section  the  thickness  is  one  half  the  width,  as  in  pul- 
leys for  transmission  by  belt.  The  width  at  the  rim 
may  be  taken  equal  to  -f  the  width  at  the  nave. 

Arms  with  flanged  cross-sections  are  generally 
straight  (Fig.  83),  and  eight  in  number,  while  those 
having  oval  cross-sections  are  curved,  either  single,  as 
explained  in  §  14,  or  double,  as  in  Fig.  84. 

To   draw  double-curved  arms    for  cable-pulleys,  we 

•p 
begin  by  striking  a  circle  with  a  radius  OA  =  —  ,  then 


ARMS  AND  NAVE   OF  CABLE-PULLEYS. 


227 


take  upon  the  circle  the  lengths  AB  and  BC,  correspond- 
ing to  the  division  by  the  arms.  Draw  the  arc  OR 
representing  one  portion  of  the  double  curve,  in  the 
same  manner  as  for  single-curved  arms.  Through  the 
centre  of  curvature  C  of  this  arc  (which,  for  eight 
arms,  is  on  the  circumference  ABC)  draw  the  line  CED, 


FIG.  83. 


FIG.  : 


and  taking  ED  —  EC,  obtain  the  radius  of  curvature 
corresponding  to  the  part  EF  oi  the  arm.  To  draw 
'  the  curves  which  limit  the  profile,  it  is  necessary  only  to 
follow  the  method  of  §  14,  remarking  that  the  centres 
for  the  arcs  are  found  upon  the  line  CD. 

When  straight  arms  are  used  the  nave  is  sometimes 
cast  with  grooves,  into  which  iron  rings  are  afterwards 
placed  ;  by  putting  on  the  rings  hot,  and  allowing  them 
to  cool,  they  are  very  firmly  fixed,  and  add  greatly  to 


228  KELTS  AND    PULLEYS. 

the  strength  of  the  pulley.  The  dimensions  of  the 
nave  are  determined,  as  already  explained  for  pulleys 
for  transmission  by  belt,  in  §  13. 

Example. — In  a  transmission  by  cable  the  radius  of 
the  pulleys  is  50  inches,  the  diameter  of  the  arbor  is 
4.8  inches,  and  that  of  the  cable  0.48  inch ;  it  is  re- 
quired to  determine  dimensions  of  the  pulley.  From 

formula  (649)  the  number  of  arms  is  A  =  4  -f-  - 

40  0.40 

=  7.  The  width  of  the  arms  at  the  nave  is,  from 
formula  (650),  A  =  4  X  0.48  -f  -  =  1.92  -f-  1.8  = 

3.72  inches.  Formula  (604),  in  which  d  represents  the 
diameter  of  the  arbor,  gives  for  the  thickness  of  the 

4.8       50 
nave  w  =  0.4  -f-  ~^  +  7"  =  °-4  +  O.8  +  I  =  2-2  inches. 

The  length  of  the  nave  (L)  ought  to  be  at  least  equal 
to  2-J  X  2.2  —  5.5  inches. 

For  very  important  transmissions  it  is  prudent  to 
have  a  reserve  cable  ;  that  is,  to  divide  the  force  to  be 
transmitted  between  two  cables,  each  having  sufficient 
strength  to  transmit  the  whole  force.  An  arrangement 
of  this  kind  is  in  use  at  SchafThouse,  in  a  transmission 
by  metallic  cable  of  600  horse-power,  of  which  we  shall 
have  occasion  to  speak  farther  on.  In  this  transmis- 
sion the  two  pulleys  are  placed  upon  one  driving  arbor, 
as  shown  in  Fig.  85.  The  pulleys  which  run  loosely 
upon  the  arbor  are  fixed  to  the  two  gear-wheels  B  and 
D,  which  engage  with  the  intermediate  gears  A  and  C. 
The  latter  gears  run  loosely  upon  their  journals,  which 
form  a  part  of  and  rotate  with  the  driving  arbor.  By 
means  of  this  arrangement  each  cable  is  made  to  trans- 


ARMS  AND  NAVE   OF  CABLE-PULLl 


mit  an  equal  share  of  the  total  force.  If  one  of  the 
cables  breaks,  the  pulley  over  which  it  ran  is  free  to 
rotate  in  the  opposite  direction,  and  the  gears  are  thus 
put  in  motion.  In  order  to  prevent  too  rapid  motion 
in  the  pulley,  which  by  the  breaking  of  a  cable  may  be- 
come loose  upon  the  arbor,  the  transmission  at  Schaff- 
house  is  provided  with  a  powerful  brake,  by  means 
of  which  the  motion  of  the  motive  turbine-wheel  may 


FIG.  85. 

be  almost  instantaneously  arrested.  Instead  of  the 
intermediate  gears  A  and  C,  simple  sectors,  such  as  are 
represented  in  the  figure  on  the  right,  might  be  used  in 
this  trrnsmission.  In  this  case  as  soon  as  a  break  in 
one  of  the  cables  occurred,  the  sectors  would  be  put  in 
motion,  and  when  the  toothless  parts  came  opposite 
the  gears  D  and  B  the  motion  of  the  pulleys  would  be 
stopped,  and  danger  of  further  accident  avoided. 


230 


BELTS  AND  PULLEYS. 


§  28.     Pulley-Supports  and  Intermediate  Pulleys 

When  the  principal  pulleys  of  a  transmission  by  cable 
are  placed  far  apart,  and  especially  when  they  are  not 
high  above  the  ground,  it  is  often  necessary  to  support 
the  cable  by  other  pulleys.  In  certain  cases  it  is  suffi- 
cient to  support  at  a  single  point  the  driven  part  of 
the  cable  while  the  driving  part  is  left  free,  as  shown  in 


FIG.  86. 


Fig.  86.  When  several  pulley-supports  are  necessary, 
the  driving  part  is  also  supplied  with  at  least  one,  as 
shown  in  Fig.  87.  In  other  cases  the  number  of  pul- 
ley-supports is  the  same  for  both  parts  of  the  cable ;  it 


FIG.  87. 


is  then  best  to  place  the  pulleys  of  the  driving  part 
directly  over  those  of  the  driven  part,  instead  of  juxta- 
positing  them,  as  has  been  several  times  attempted,  and 
which  causes  rapid  wear  of  the  cable,  consequently 
produces  a  wearing  friction  upon  the  pulley-grooves, 
and  also  tends  to  make  the  cable  run  off  the  pulleys. 


PULLEY-SUPPORTS—INTERMEDIATE  PULLEYS.    2$l 

In  the  arrangement  represented  in  Fig.  88  the  pulley- 
supports  of  the  driving  part  are  placed  under  those  of 
the  driven  part  in  order  to  gain  space  above  the  ground. 


FIG. 


In  most  cases  when  the  distance  between  the  princi- 
pal pulleys  makes  a  great  number  of  pulley-supports 
necessary,  this  arrangement  may  be  advantageously 
replaced  by  a  series  of  successive  transmissions  (Ziegler), 
Fig.  89.  The  pulley-supports  of  Fig.  88  are  then  re- 
placed by  intermediate  double-grooved  pulleys  placed 
at  as  near  the  same  distances  apart  as  possible,  so  that 


M, 


FIG.  89. 

in  case  of  breakage  in  any  of  the  cables  a  single  reserve 
cable  may  be  used  to  replace  it.* 

*This  has  been  done  by  Ziegler  at  Frankfort-on-the-Main,  where  a 
force  of  100  horse  power  is  transmitted  at  a  distance  of  984  metres — 
nearly  £  of  a  mile. 


232 


BELTS  AND  PULLEYS. 


The  different  points  at  which  a  cable  is  supported 
are  called  stations ;  those  which  correspond  to  the 
principal  pulleys  of  the  transmission  are  called  the  sta- 
tions at  the  extremities  and  the  others  intermediate 


FIG.  90. 

stations.  Sometimes  it  is  necessary  to  change  the 
directions  of  the  cable  at  an  intermediate  station  ;  Him 
has  proposed  to  accomplish  this  change  of  direction  by 
means  of  a  horizontal  pulley,  Fig.  90,  while  it  has  also 
been  suggested  to  use  a  pair  of  bevel  gears,  Fig.  91. 


FIG.  91. 

The  use  of  transmissions  by  cable  is  very  convenient 
when  we  wish  to  divide  between  several  establishments, 
belonging  to  different  proprietors,  the  force  derived 
from  a  single  motor :  to  do  this  we  have  simply  to 


PULLE  Y-SUPPORTS— INTERMEDIA  TE  PULLE  VS.    233 

make  the  intermediate  stations  the  starting-points  or 
stations  at  one  extremity  of  supplementary  transmis- 
sions. Stations  of  this  kind  are  called  division-stations. 
Pulley-supports  are  also  used  in  the  special  case  in 
which  the  driven  arbor  is  placed  almost  vertically  above 
or  below  the  driving-arbor.  There  would  be  serious 
difficulty  in  making  use  of  an  inclined  cable,  connect- 
ing directly  the  two  pulleys  A  and  B,  Figs.  92  and  93 ; 
it  is  preferable  by  far  to  use  the  pulley-supports  7",  T, 
placed  in  such  a  manner  that  one  part  of  the  cable,  TA 
or  TB,  may  be  horizontal.  It  is  then  sufficient  to 


determine,  by  means  of  the  preceding  rules,  the  proper 
tensions  to  give  to  the  horizontal  part  of  the  transmis- 
sion without  reference  to  the  inclined  part. 

The  use  of  cables  for  the  transmission  of  forces  to 
great  depths — into  the  shafts  of  mines,  for  example — is 
still  in  a  period  of  development.  We  may  say,  how- 
ever, from  attempts  already  made  in  this  direction,  that 
satisfactory  results  have  been  obtained.* 

*  Review  of  Society  of  German  Engineers,  1866,  p.  371.     Werner, 
"  Use  of  transmissions  by  metallic  cables  for  the  shafts  of  mines." 


234 


BELl'S  AND  PULLEYS. 


We  meet  with  a  remarkable  example  of  this  mode  of 
transmission  in  the  arrangement  at  Schaffhouse,  where 
a  force  of  about  600  horse-power,  taken  from  the  current 
of  the  Rhine,  is  received  by  turbines  at  the  left  bank, 
and  is  intended  to  be  transmitted  across  the  river  to  the 
right  bank,  there  to  be  divided  among  several  factories. 
This  important  application,  credit  for  which  is  due  to 
the  Society  of  Hydraulic  Engineers  of  Schaffhouse,  is 
very  nearly  completed,  and  affords,  in  all  its  details, 
information  of  the  greatest  interest  to  engineers. 

§  29.     Dimensions  of  Pulley-supports. 

The  pulleys  intended  to  support  the  driving  part  of 
the  cable  ought  properly  to  have  the  same  diameter  as 
the  pulleys  of  transmission ;  those  supporting  the 
driven  part  may,  in  normal  transmissions,  have  smaller 
dimensions.  The  following  table  indicates  the  limits 
below  which  we  should  not  take  the  radius  RQ  of  the 
pulley-supports. 

The  numbers  contained  in  the  table  have  been  cal- 
culated by  means  of  the  formula 

R0  _      28446000 
6  ""  51202.8  —  S\' 


Si 

J 

*0 

8 

^i 

j 

*o 
8- 

711-15 

24890.25 

563 

12800.70 

12800.70 

741 

1422.30 

24179.10 

571 

14223.00 

11378.40 

769 

2844.60 

22756.80 

588 

15645.30 

9956.10 

800 

4266.90 

21334.50 

606 

17067.60 

8533.80 

833 

5689.20 

19912.20 

625 

18489.90 

7HI.50 

870 

7111.50 

18489.90 

645 

19912.20 

5689.20 

909 

8533.80 

17067.60 

667 

21334.50 

4266.90 

952 

9956.10 

15645.30 

690 

22756.80 

2844.60 

1000 

11378.40 

14223.00 

714 

24197.10 

1422.30 

1053 

PRESSURE   ON  PULLEY-SUPPORT  AXES.        235 

The  values  contained  in  the  table  furnish  excellent 
dimensions  for  ^0  principally  for  large  values  of  Sr  In 
transmissions  with  increased  tension  (see  §  21)  the 
difference  between  RQ  and  R  is  so  small  that  we  may 
take,  without  disadvantage,  R  =  R0.  In  compound 
transmssions  (see  §  28)  there  is  no  difference  in  size  be- 
tween the  principal  pulleys  at  the  extremities  and  the 
intermediate  pulleys. 

§  30.     Pressure  upon  the  Axes  of  Pulley-supports. 

In  a  transmission  by  cables,  which  we  have  taken 
care  to  calculate  for  its  entire  length,  we  should  know 
the  tensions  at  each  station,  and  (from  the  curves  of 


FIG.  94. 


FIG.  95. 


the  cables  traced  according  to  §  23)  the  directions  of 
the  different  parts  which  are  to  be  supported  by  in- 
termediate pulleys.  For  example,  in  Fig.  94,  for  an 
intermediate  pulley  we  should  know  the  values  of  T, 


236  BELTS  AND  PULLEYS. 

t,  TV  and  ^  and  their  directions.  We  can  then  deter- 
mine by  means  of  formulas  already  given  the  approxi- 
mate weight  of  the  pulley,  which  allows  us  to  trace 
graphically  (Fig.  95)  the  resultant  Q  of  the  different 
forces.  To  accomplish  this  we  draw  the  lines  A  B, 
BC,  CD,  D  E,  and  EF  respectively  equal  and  parallel 
to  T,  TV  t,  /„  and  G.  The  line  A  F,  which  completes 
the  polygon,  represents  in  amount  and  direction  the  re- 
sultant Q. 

Pulley-supports  are  ordinarily  in  construction  iden- 
tical with  the  principal  pulleys  for  the  same  diameter 
of  cable.  By  virtue  of  the  rules  of  §§  26  and  27,  the 
following  formulas  may  be  obtained  for  the  approxi- 
mate weights  of  the  pulleys  : 

For  single-grooved  pulleys, 

G  ry       .   145-6  .   ii5.52\/M   ,    / 

51  =  0.034375  [(45  +  -^-  +  -JS-J  bJ  +  (0.33  + 


For  double-grooved  pulleys, 

G  VI       ,   265.6  ,         .  , 

f  =  0.034375  L(84  +  -jr  +  -y-         +  0.33  + 

0.464  ,  o.n5\  [RY  ,    /  ,  o.oo28W^V 

•    (653) 


Example. — In  the  fourth  example  of  §  20  for  a  radius 
of  30  inches  the  diameter  of  the  wires  (of  which  there 
are  36)  is  0.036  inch.  The  diameter  of  the  cable 
itself  is  therefore  d  —  8  X  0.036  =  0.288  inch,  which 


PRESSURE   ON  PULLEY-SUPPORT  AXES.         237 

gives    ,  —  — ^TO  —  !O4-     The  weight  of  the  pulley  for 
a  single  groove  is,  from   formula  (652),  £=0.024  X 


FIG.  96. 

D  145.6  115. 52\  /  —r— r 

45  +  0^88  +  ^083-]  I04  +  l°'33+  o^« 

^  1 1  <;\  /  ,   o.ooaSN  ~| 

Sfj  10816  +  (0005  +  ^^^^-J  1124864]  = 


2°4 


pounds. 


238  BELTS  AND  PULLEYS. 

Example. — For  the  transmission  of  300  horse-powei 
of  the  second  example  of  §  20  we  have  d  —  0.087, 
which  for  a  cable  of  60  wires  gives  </=  12.8  X  0.087  — 

D  O  - 

1. 1 1  inches,  R  =  85  inches.     Consequently  -3  = =: 

77.     The  weight  of  the  pulley  for  double  grooves  is 

rt       265.5     212. 8\ 

therefore  G  =  1.37  X  o.O34375[_V84  +  ^Tf  +  ^yj 

/          ,   0.464       o.iis\  /  .    o.oo28\ 

77  +  10.33  +  -  5929  +    0.005  + 

\    °°        i. ii          1.23  /-^  ^       \  i. ii  / 


i 


456S33J  =  2I93  pounds. 

For  very  large  transmission-pulleys  the  weights  be- 
come important  considerations,  as  may  be  seen  by  the 
last  example.  For  this  reason  engineers  have  sought, 
by  modifying  the  system  of  construction,  to  reduce  the 
weights.  By  adopting  the  arrangement  represented  in 
Fig.  96,  in  which  the  arms  are  formed  by  two  series  of 
inclined  rods  meeting  in  pairs  at  the  rim  of  the  pulley, 
the  weights  may  be  reduced  to  about  three  quarters 
those  given  by  the  preceding  formulas.  In  Sweden, 
where  transmission  by  cable  has  already  been  firmly 
established,  pulleys  constructed  of  sheet -iron  have 
been  successfully  employed.* 

§  31.     Station  Pillars. 

Fig.  96  represents  the  arrangement  of  a  station  for4 
the  intermediate  pulleys  of  a  compound  transmission. 
To  support  the  pedestals  for  the  axle  of  a  pulley  of 
this  kind  we  may  with  propriety  build  up  a  frame-work 

*See  Annals  of  the  Society  of  German  Engineers,  1868,  p.  5Qi. 


STATION  PILLARS. 


239 


of  wood  ;  it  is,  however,  preferable  to  use  a  solid  pier 
of  brick  or  stone,  upon  which  are  fixed  either  low 
pedestals,  as  in  the  figure,  or  high  pedestals,  such  as 


FIG.  97. 


Figs.  97  or  98,  which  are  especially  advantageous  when 

the  height  of  the  pulley-axes  above  the  ground  is  great. 

The  pedestal-plates  are  fastened  to  the  pier  by  means 


240 


BELTS  AND  PULLEYS. 


FIG.  08. 


STA  TION  PILLARS. 


241 


of  four  strong  anchor-bolts  passing  through  the  pier 
and  into  the  foundation.  The  length  of  the  axle  be- 
tween the  centres  of  the  journals  is  generally  taken 
equal  to  the  radius  of  the  pulley.  In  stations  for  two 
pulleys  the  pier  is  divided  to  a  greater  depth,  and  the 
axle  of  the  upper  pulley  is  supported  by  high  pedes- 
tals. In  certain  cases  the  two  pulleys  are  placed  side 
by  side,  as  indicated  by  the  dotted  lines  ir  Fig.  96— 


FIG.  99. 

an  arrangement  especially  convenient  for  putting  on 
.  the  cable.  Because  of  the  weight  of  metallic  cables 
this  operation  is  by  no  means  simple ;  to  accomplish  it 
Ziegler  has  employed  an  arrangement  similar  to  Her- 
land's  tool  for  putting  on  belts.  Fig.  99  represents 
the  arrangement,  which  consists  of  a  curved  piece  of 
angle-iron,  fixed  in  the  groove  of  one  of  the  pulleys  by 
means  of  hooked  bolts  (see  figure  in  centre).  In  the 
left-hand  figure  the  cable  is  at  the  side  of  the  pulley; 


242  BELTS  AND  PULLEYS. 

in  the  right-hand  figure  it  rests  in  the  groove  of  the 
pulley. 

Although  throughout  this  entire  chapter  we  have 
assumed  that  the  two  pulleys  of  transmission  have  the 
same  diameter,  it  does  not  follow  that  we  may  not  use 
transmission-pulleys  of  different  diameters.  Indeed  it 
may  sometimes  be  necessary  to  have  such  an  inequal- 
ity of  pulleys.  In  all  cases  of  this  kind  it  is  best  to 
confine  ourselves  to  the  determination  of  the  dimen- 
sions of  the  smaller  pulley  and  the  corresponding 
diameter  of  the  cable;  taking  care,  however,  not  to 
lose  sight  of  the  fact  that,  in  order  to  obtain  the  best 
results  from  our  transmission,  it  is  essential,  first  of  all, 
that  the  diameters  of  our  pulleys  be  no  smaller  than 
the  limits  indicated  in  the  preceding  pages. 


APPENDIX. 


i. 

ACTUATED  by  a  desire  to  obtain,  by  experiment  with 
the  belts  and  pulleys  in  ordinary  practical  use,  the  co- 
efficient of  friction  which  should  be  used  in  belt-calcu- 
lations, the  author  provided  himself  with  apparatus, 
and,  before  making  use  of  the  coefficient  value  <p  —  0.40 
in  this  work,  very  carefully  proved  this  value  as  the 
mean  of  a  number  of  trials.  The  apparatus  consisted 
of  the  following  arrangement : 

Fig.  100.  A  pulley  A  securely  fastened  by  the  pins 
x,  x,  so  that  it  could  not  move  in  any  direction ;  a  belt 
£,  B  passed  around  the  pulley,  and  its  ends  attached 
to  the  levers  abc  and  a'b'c' ;  two  weights  w  =  20 
.pounds  and  W  =  40  pounds,  the  latter  being  arranged 
so  that  it  could  be  moved  along  the  lever-arm  be  at 
will.  Belts  and  pulleys  which  had  been  used  for  some 
time— not,  however,  badly  worn  or  injured — were  pur- 
posely chosen  in  order  to  obtain  more  practical  results. 

The  fulcrums  b  and  V  were  metallic  knife-edges,  and 
the  friction  between  them  and  their  levers  therefore 
practically  nothing.  The  weight  w  =  20  pounds  was 


244 


BELTS  AND  PULLEYS. 


fixed  upon  the  lever  a'b'c,  the  arms  being  a'b!  =  4 
inches  and  b'cf  =  12  inches.  The  tension  t  was  there- 

1 2 

fore,  from  the  principles  of  the  simple  lever,  20  X  — 

•=•  60  pounds.  In  each  experiment  the  arm  ab  was  4 
inches  long  and  the  weight  W  was  moved  along  the 
lever-arm  be  until  the  tension  T  was  such  that  the  belt 
was  just  on  the  point  of  slipping;  the  corresponding 


FIG.  ioo. 

arm  was  then  carefully  measured  with  an  accurate 
hundredth  rule,  and  the  tension  T  calculated  as  above 
for  /. 

Experiment  i. — The  angle  embraced  by  the  belt  was 
a  —  180  degrees,  the  tension  /  =  60,  and  the  lever-arm 
be  —  22.50  inches.  The  tension  T  was  therefore  T  = 


22.50 


X   40  =  225  pounds. 


225 
- 


APPENDIX. 


245 


log  -  =  log  3.75  —  0.57403.     From  formula  (41),  by 

transposing,  we  obtain  for  the  coefficient  of   friction 

T 

the  expression   q>  =  log  —  -=-  0.007578^,  which  in  the 

0.57403  0.57403 

present  case  becomes  V  =  ^—^-  =  -j.^ 

or  cp  =  0.42083. 

In  this  experiment,  although  tried  with  five  different 


FIG.  ioi. 

pulleys  and  as  many  different  belts,  the  greatest  value 
obtained  for  the  coefficient  was  <p  =  0.4248,  and  the 
smallest  cp  =  0.41997.  The  value  determined  above 
was  nearest  to  the  average  of  the  different  experiments. 
Experiment  2. — The  angle  embraced  by  the  belt  was 
a  =  90°.  The  arrangement  of  apparatus  for  this  ex- 
periment is  shown  in  Fig.  ioi,  being  the  same  as  in 
Experiment  i,  except  that  the  belt  was  held  away  from 


246  BELTS  AND  PULLEYS. 

the  pulley  by  means  of  two  small  rollers  7,7,  in  order 
to  obtain  the  required  value  for  the  angle  a.  The 
friction  of  the  rollers  was  so  small  that  when  used  for 
an  angle  a  =  180°,  with  the  belt  and  pulley  which 
gave  the  mean  result  in  Experiment  i,  and  with  the 
same  angle  of  deviation  (kyz.  Fig.  101),  made  a  differ- 
ence of  only  0.00197  in  the  results — practically  none  at 
all. 

The  tension  /  was  as  before  /  —  60  pounds,  and  the 
lever-arm  be  =  11.64  inches.  The  greater  tension  was 

therefore  T  =  -      -  X  40  =  116.4  pounds.     Hence  — 
116.4  T 

—  —^-  1.94,    log   -   :   :    log     1.94    =     0.28780,     <p     - 

0.28780  0.28780 

x =  — z^ or   <p   =    0.42108  —  nearest 

0.007578  X  90       0.68202 

average  of  different  experiments. 

Experiment  3. — The  angle  embraced  by  the  belt  was 
a  =  45°,  tension  /  =  60  pounds,  lever-arm  be  —  8.4 

8.4  T 

inches.    Consequently  T—  -  ~  X  40  =  84  pounds,  —  = 

84  T  0.14613 

•7—  =    1.4,  log-  —  log  1.4=  O.I46I3,  (D  —  

60  b/  0.007578X45 

O.I46I3 

—  -        —  ,or  (p  =  0.42852 — nearest  average  value. 
0.34101 

In  this  experiment  the  angle  of  deviation  (kyz,  Fig. 
101)  was  so  great  that  the  friction  of  the  rollers  y,y 
must  have  had  an  appreciable  effect,  which  probably 
accounts  for  the  increased  value  of  cp. 

Experiment  4. — The  angle  embraced  by  the  belt  was 
210° ;  the  arrangement  of  apparatus  in  order  to  obtain 


APPENDIX. 


247 


this  angle  is  shown  in  Fig.  102.    Tension  /  =  60  pounds, 
and  the  lever-arm  be  =  28.4   inches.      Therefore  the 

greater  tension  was  T  =  -    -  X  40  =  284  pounds,  and 

—  =  -£--  —  4.733.      The    logarithm    of    this    ratio    is 
0.675 14,  and  we  have,  for  the  coefficient, 

0.42425 — nearest 


=          0.67514         ^0.67514 
0.007578X210  "     1.59138 


average  value. 

Example  5. — The  angle  embraced  by  the  belt  was 


c' 


FIG.  102. 


a  —  250°,  tension  /  =  60  pounds,  and  the   lever-arm 
be  39.15  inches  long.     The  greater  tension  was  there- 


fore r  =  32l5  X 


=  391.5  pounds,    r  = 


248 


BELTS  AND  PULLEYS. 


0.81458 
- 


6.525,  log-  =  0.81458.     Hence  v  = 

0.81458 

—  —  -  -  =  0.42997  —  nearest  average  value. 
1.5945 

In  each  of  these  experiments  five  trials  were  made 

with  different  belts  and  pul- 
leys, the  values  worked  out 
above  being  about  mean  for 
each  separate  experiment, 
A  mean  between  the  five 
values  given  above  is  there- 
fore a  mean  value  deter- 
mined by  twenty-five  very 
careful  experiments,  and 
may  be  relied  upon  for 

t  ^^     practical  calculations.    This 

FlG-  I03-  T      gives  for  us  our  coefficient  of 

friction  between  leather  belts  and  iron  pulleys  *  cp  = 
0.42507.  Since  in  this  coefficient  there  is  not  the  same 
need  of  a  factor  of  safety  as  in  calculations  with  proof- 
strengths  and  to  prevent  breakage,  we  may  take  very 
nearly  the  full  value  without  running  risk  of  any  serious 
accident.  We  have  taken,  and  shall  use  throughout 
this  work,  the  value 

cp  =  0.40. 

All  the  belts  with  which  the  above  experiments  were 
made  had  been  oiled  to  a  moderate  extent  with  castor- 
oil. 

*This  value  practically  agrees  with  the  results  of  the  experiments 
of  Messrs.  Briggs  and  Tovvne,  as  given  in  Journal  of  the  Franklin  In- 
stitute, January,  1868. 


APPENDIX.  249 

A  series  of  18  experiments  with  new  dry  leather  belts 
hung  over  a  fixed  pulley  and  weighted  at  each  end 
(see  Fig.  103)  gave  an  average  value  of 

<p  —  0.304. 

The  angle  embraced  by  the  belt  in  each  case  was 
180°,  the  weights  on  the  ends  varying  from  10  and  25^ 
pounds  to  90  and  229  pounds. 

The  author  also  tried  21  experiments  with  some  old, 
gummy  leather  belting  which  had  lain  in  a  dry  room 
for  nearly  two  years,  and  to  his  astonishment  found  an 
average  value  of  q>  —  0.61  for  the  coefficient  of  fric- 
tion. These  belts,  which  were  2  inches  wide  and  T3^ 
inch  thick,  broke  through  the  solid  parts  when  tested 
for  strength,  at  an  average  strain  of  1088  pounds.  This 

o 

would  give  for  the  ultimate  strength  1088  X  —,  or  about 

2900  pounds  per  square  inch — very  little  if  any  below 
that  of  ordinary  belt-leather. 

Leather  over  Leather-covered  Pulleys. — Using  the 
belting  and  pulleys  of  the  first  five  experiments  men- 
tioned in  this  Appendix,  the  author  tried  the  following 
experiments  with  leather-covered  pulleys : 

Experiment  i. — With  apparatus  of  Fig.  100.  a  = 
1 80°,  t  =  60  pounds.  The  lever-arm  be  was  26.15 
inches  long  when  the  belt  began  to  slip.  Hence  T  = 

26.15  T  261.5 

— —  X  40  =  261.5  pounds,  log  -  ==  log  -^-  =  log 

4.358  =  0.63929.    Consequently  <p  —  - 


'0.007578  X  1 80 
0.63929 

--"  =  0.4687. 
1.36404  ' 


250  BELTS  AND  PULLEYS. 

Experiment  2. — With  apparatus  of  Fig0   101.     a  = 
90,  /  =  60  pounds.     The  lever-arm  be  was  12.61  inches 

long.     Hence   T  —  -       '-  X  40  =  126.1,  log  -  =  log 

4  * 

126.1  0.32263 

=  log  2.102  =  0.32263,  and  <p  =  — 


60  -0.007578X90 

0.32263 

=  °473' 


Experiment  3.  —  With  apparatus  of  Fig.   101.     a  = 
45°,  ^  —  60  pounds.     The  lever-arm  fc  was  8.75  inches 

Q    yC  7^ 

long.    Hence  7"=  —  —  X  40  =  87.5  pounds,  log  —  =  log 

4  * 

87.5  0.16376 

—  Jog  J.458  —  0.16376,  and  cp       -  - 


.  ., 

60  0.007578  X  45 

0.16376 

---  ±L-  =  0.4802. 

0.34101 

Experiment  4.  —  With  apparatus  of  Fig.   102.     a  = 
210°,  /  =  60*  pounds.     The   lever-arm   be  was   34.99 

34.00 
inches  long.     Hence  T  =  -       -  X  40  =  349-9  pounds, 


log  -  =  log  =  log   5.832  =  0.76582,  and   cp  = 

0.76582  0.76582 


0.007578  X  210    "  I .$9138 

Experiment  5. — With  apparatus  of  Fig.  102.     a  = 
250°,  t  =  60  pounds.      The   lever-arm   be  was  49.77 

inches  long.      Consequently  T  =  -       -  X  40  —  497-7 

pounds,  log  -  =  log     ^     —  log  8.295  =  0.91882,  and 
0.91882  0.91882 

CD  = —    — —   0.485. 

0.007578  X  250         1.8945 


APPENDIX.  N^gi  /£          2  5  1 

Each  of  the  above  experiments  is  the  one  giving  the 
nearest  average  value  of  cp  out  of  five  tests.  The  aver- 
age of  these  five  experiments  is  therefore  the  average 
of  twenty-five  carefully-made  trials.  This  average 
value  is  cp  =  0.4776,  which  permits  us,  after  making 
fair  allowances  for  the  friction  of  the  rollers  y,  y  used 
in  the  apparatus,  to  use,  for  the  coefficient  of  friction 
of  leather  over  leather-covered  pulleys,  the  value 

<p  =  0.45. 

Experiments  tried  with  new  dry  leather  over  pulleys 
covered  with  the  same  gave  for  the  coefficient  of 
friction 

cp  —  0.348. 

Vulcani  zed-rubber  Belts.  —  The  author  has  made 
nearly  sixty  different  trials  with  vulcanized-rubber 
belts  over  cast-iron  pulleys,  the  belts  and  pulleys  hav- 
ing been  used  to  a  moderate  extent  for  practical  pur- 
poses ;  the  mean  value  for  the  ratio  of  the  tensions  for 
a  =  1  80°  was  found  to  be 

T 

1  =  4-55. 

From  this  we  have 

T 

log-  =  log  4.5  5  =  .0658011, 

and  consequently 

0.658011          _  0.658011 

"  ~ 


0.007578  X  1  80  ~     1.36404" 


252  BELTS  AND  PULLEYS. 

This  value  is  slightly  greater  than  that  obtained  for 
oiled  leather  belts  over  leather-covered  pulleys ;  for 
reasons  given  in  §  12,  however,  we  take  the  coefficient 
of  friction  for  vulcanized-rubber  belts  over  cast-iron 
pulleys^  the  same  as  for  leather  over  leather-covered 
pulleys,  i.e., 

<p  =  0.45. 


II. 

THE  principles  of  the  endless  belt  and  pulleys,  as  ap- 
plied for  numerous  different  purposes  of  the  shops  and 
factories,  have  developed  from  time  to  time  a  vast 
number  of  ingenious  contrivances  by  means  of  which 
many  motions  other  than  that  of  simple,  continuous 
rotation  may  be  easily  obtained.  Thus,  while  the 
fundamental  mechanism  itself  has  remained  essentially 
unchanged  from  the  forgotten  ages  which  gave  it  birth 
to  the  present'time,  instead  of  the  simple  band  trans- 
mitting the  motion  of  one  pulley  to  another  parallel  and 
equal  pulley,  we  have  now  devices  by  means  of  which 
we  may  transmit  the  motion  of  the  driver  to  one  or  more 
shafts  oblique  in  almost"  any  direction  to  that  of  the 
motive  pulley ;  to  increase  and  decrease  the  speeds  of 
the  various  pulleys  at  will ;  to  reverse  and  change 
the  directions  of  rotation  of  any  or  all  of  the  driven 
pulleys ;  to  almost  instantly  impart  or  arrest  their 
motions ;  and  to  transform  the  continuous,  rotary 
motion  of  the  driver  into  rectilinear,  reciprocating,  in- 
termittent, and  intricate  compound  motions.  Indeed 
it  is  not  so  far  from  the  truth  as  is  sometimes  sus- 


APPENDIX. 


253 


pected,  that  "  almost  anything  can  be  accomplished 
with  a  room  full  of  pulleys  and  belts." 

Some  of  the  most  common  of  the  many  devices 
made  use  of  to  obtain  the  different  motions  necessary 
for  the  various  kinds  of  work  known  to  the  artisan 
have  been  gathered  together  and  explained,  as  briefly 
as  is  consistent  with  a  clear  understanding  of  the 
mechanisms,  in  the  following  pages. 

Fig.  104  represents  an  arrangement  of  pulleys  by 
means  of  which  the  motion  of  the  driving-pulley  A 


FIG.  104. 

may  be  transmitted  to  the  pulley  B,  at  right  angles 
with  the  driver.  The  guides  C,  C  hold  the  band  in 
such  positions  that  it  runs  easily  upon  the  driven 
pulley,  about  which  it  makes  one  entire  turn. 

Fig.  105  represents  a  device  for  transforming  a  rec- 
tilinear into  a  reciprocating  rotary  or  oscillating  mo- 
tion. A  rocking  motion  is  given  to  the  arm  E  by 
means  of  the  connecting-rod  A  and  the  motive-piece 


254 


BELTS  AND  PULLEYS. 


F.  The  required  motion  is  then  transmitted  to  the 
two  pulleys  C  and  D  by  means  of  the  half-pulley  B  and 
the  belt  gg.  Inversely,  by  giving  an  oscillating  motion 
to  one  of  the  pulleys  C  and  D  a  reciprocating  recti- 
linear motion  may  be  given  to  the  piece  F. 

In  the  combination  of  pulleys  represented  in  Fig. 
106,  the  uniform  rotary  motion  of  the  driver  A  gives 
to  the  pulleys  B,  C,  and  D  uniform  rotary  motions  at 
different  speeds,  according  to  their  diameters,  as  ex- 


FIG.  105. 


FIG.  106. 


plained  in  §2.  The  driven  pulleys  are  in  different 
planes,  and  the  face-width  of  the  driver  is  great  enough 
to  carry  the  several  belts  without  interfering  with  each 
other. 

The  contrivance  shown  in  Fig.  107  is  intended  to 
give  two  rotary  motions  in  contrary  directions  to  two 
coincident  shafts.  The  pulley  A  is  the  driver,  and 
carries  a  crossed  belt  running  to  the  pulley  B,  and  also 
an  open  belt  running  to  the  pulley  C.  The  latter  pul- 
leys consequently  rotate  in  opposite  directions  at  the 


APPENDIX. 


same  or  at  different  speeds,  according  as  their  diameters 
are  equal  or  different. 

Fig.  108  represents  an  arrangement  by  means  of 
which  a  great  increase  of  friction  between  the  band 
and  principal  pulleys  may  be  obtained.  The  band 
passes  several  times  around  the  principal  pulleys,  A  and 
B,  and  is  properly  guided  by  means  of  the  small  rollers 
b  and  c,  as  shown  in  the  figure.  Let  us  suppose  that 


FIG.  107. 


FIG.  108. 


the  band  passes  four  times  around  the  pulleys  :  the 
arc  embraced  is  then  a  =  360  X  4  =  1440°.  If  we 
take  for  the  coefficient  of  friction  cp  =  0.30,  we  shall 
have  for  the  ratio  of  the  tensions  [see  formula  (41)], 

T 

log  -  =  0.007578  X  0.30  X  1440  =  3-2737. 


or 


•T 

-  =  1878. 


256 


BELTS  AND  PULLEYS. 


If  the  band  passed  only  ^  times  around  the  pulleys, 
we  should  have  a  =  180°,  and  consequently 


log-  =  0.007578  X  0.30  X  180  =  0.4092, 


or 


j  =  2.566. 


In  other  words,  by  means  of  the  above  arrangement 
the  ratio  of  the  greater  to  the  smaller  tension  is  in- 
creased over  seven  hundred-fold. 


An  ingenious  device  for  transmitting  a  rotary  mo- 
tion to  a  movable  pulley  by  means  of  an  ordinary  belt 
is  represented  in  Fig.  109.  A  is  the  driver  and  B  the 
movable  driven  pulley.  The  intermediate  pulley  C  is 
suspended  by  means  of  a  cord  passing  over  the  roller 
D  and  the  weight  W\  it  is  thus  free  to  move  up  and 
down  in  the  frame  F,  and  the  pulley  Z?  may  be  moved 
through  considerable  distances  without  interfering  with 


APPENDIX. 


257 


the  motion  of  the  belt,  as  is  indicated  by  the  dotted 
lines. 

Fig.  no  represents  an  arrangement  of  pulleys  for 
transmitting  two  different  speeds  from  the  driving- 
shaft  AA 
shaft  BB. 


to  the  driven 
Each  shaft  car- 
ries two  fast  pulleys  (C9  C, 
C,  and  £)  and  two  loose 
pulleys  (F,  F,  F ,  and  G\ 
and  the  two  belts  xx  and 
yy  are  moved  together, 
backward  and  forward, 
across  the  pulley-faces. 
When  the  belts  are  in  the 
positions  shown  in  the  fig- 


C     F 


F'  c' 


X 


X 


y 


y 


C     F 


G     E 


FIG.  no. 

ure,  the  belt  xx  is  at  work  and  yy  at  rest.  When  the 
belts  are  moved  to  the  pulleys  F,  F  and  C,  E,  the  belt 
xx  is  at  rest  and  the  belt  yy  transmits  to  the  driven 
shaft  a  fast  motion  because  of  the  small  diameters  of 
the  pulleys  G  and  E. 


FIG. 


A  mode  of  transforming  a  reciprocating  rectilinear 
motion  into  an  alternating  rotary  motion  is  shown  in 
Fig.  in.  To  the  piece  A  A  a  cord,  which  also  passes 


258 


BELTS  AND  PULLEYS. 


around  the  pulley  C,  is  attached,  and  by  moving  the 
piece  AA  backward  and  forward,  the  required  motion 
of  C  is  obtained.  Inversely,  by  giving  an  oscillating 
motion  to  the  pulley  C,  a  reciprocating  rectilinear  mo- 
tion may  be  given  to  the  piece  A  A. 


Fig.  112  represents  an  arrangement  by  means  of 
which  the  continuous  rotary  motion  of  the  driver  C 
may  be  transmitted  to  two  pulleys  A,  A  at  right  an- 
gles with  the  motive-pulley.  A  portion  of  the  band  is 
horizontal,  and  runs  without  difficulty  upon  the  pulleys 
Ay  A  ;  the  other  part  is  guided  by  the  intermediate 
pulleys  B,  By  as  shown  in  the  figure. 


W 


FIG.  113. 

Fig.  113  represents  a  device  by  means  of  which  a 
variable  motion  for  the  driven  shaft  may  be  obtained 
from  the  uniform  motion  of  the  driver.  The  pulley  A 
is  the  driver,  and  bears  upon  its  face  a  deep  groove,  as 
shown  by  the  dotted  circle.  The  pulley  B  is  mutilated, 
one  part  having  a  greater  radius  than  the  other,  and 


APPENDIX. 


259 


also  has  a  deeply  grooved  face.  When  the  pulleys  are 
in  the  positions  shown  in  the  figure,  the  belt  is  tight, 
and  drives  uniformly  the  pulley  B.  When,  however, 
the  smaller  part  of  the  pulley  B  comes  opposite  the 
belt,  the  tension  is  greatly  lessened  and  the  weight  W 
jerks  the  pulley  quickly  around  until  the  belt  again  be- 
comes tight,  when  the  operation  is  repeated. 

In  Fig.  1 14,  the  arm  C  bearing  the  pulley  JB,  when 
moved  up  and  down  in  treadle 
motion,  transmits  a  rotary  mo- 
tion to  the  shaft  5  by  means 
of  the  belt  and  eccentric  pul- 
ley A.  By  making  the  pulley 
A  the  driver,  the  pulley  B  may 
be  given  a  rotary  motion,  and 
at  the  same  time  an  oscillating 
motion  about  the  point  D. 

Fig.  1 1 5  represents  a  device 
for  transmitting  a  continuous  rotary  motion  to  a  mov- 
able shaft.     The  pulley  A  is  the  driver,  and  the  driven 

pulley  B  may  be  moved 
about  in  the  frame  C,  as 
shown  by  the  dotted  lines, 
without  interfering  with  the 
motion  of  the  belt.  The 
radius  of  curvature  of  the 
axis  of  the  frame  is  equal 
to  the  distance  between  the 
centres  of  the  pulleys. 

Another  mode  of  trans- 
mitting a  rotary  motion  to 
a  movable  pulley  is  shown  in  Fig.  116.     A  is  the  driv- 


FIG.  114. 


FIG.  115. 


260 


BELTS  AND  PULLEYS. 


FIG.  116. 


ing-pulley  and  carries  an  elastic   belt   of  india-rubber. 
By  stretching   the   belt,    the    driven 
\A    pulley  B  may  be  moved  about  in  al- 
'      most   any  direction,  as  indicated  by 
the  dotted  lines.     This  device  is  used 
extensively  in  hair-cutting  and  clip- 
ping machines,  and  dental  apparatus 
for  boring  and  drilling. 

Fig.  117  represents  a  device  known 
to  artisans  as  the  "  frictionless  bear- 
ing," or  "  anti-friction  bearing.  The  shaft  b  of  the  pul- 
ley^, instead  of  turning  in  an  ordinary  pedestal  or 
hanger,  rests  upon  the  circum- 
ferences of  six  small  rollers,  c, 
c,  etc.  The  friction  due  to  the 
weight  of  the  pulley  and  shaft 
is  thus  distributed  among  the 
six  rollers,  and,  since  the  shaft 
rolls  upon  the  rollers  instead  of 
sliding  around  in  the  pedestal 
as  with  common  bearings,  the 
friction  of  sliding  is  eliminated.  FIG.  n7. 

Considerable  difference  of  opinion  exists  among  me- 
chanical men  as  to  the  best  method  of  connecting  the 
various  shafts  in  shops  and  mills  with  the  driving 
drum  or  pulley.  Some  engineers  claim  that  but  one 
belt  should  be  used  to  drive  all  the  shafts  in  the  mill ; 
that  this  method  is  the  most  advantageous,  because  of 
the  great  duration  of  the  driving-belt  and  because  of 
the  simplicity  of  the  arrangement.  Others  suggest  two 
belts — one  connecting  the  driving-pulley  with  the  first 
shop-shaft,  and  the  other  passing  from  the  first  shop- 


APPENDIX. 


26l 


shaft  to  all  the  other  shafts  ;  while  by  many  it  is  claimed 
that  each  principal  shaft  should  have  its  own  belt 
connecting  it,  either  directly  or  indirectly,  with  the 
driving-pulley. 


FIG. 


Fig.  118  represents  a  section  of  a  three-story  mill, 
the  shafts  of  which  are  driven  by  means  of  a  single 
belt.  The  arrangement  of  the  various  pulleys  is  suffi- 


262  BELTS  AND  PULLEYS. 

ciently  clear  in  the  figure  without  further  explanation. 
The  objections  offered  to  this  method  of  transmitting 
to  the  different  shafts  the  power  of  the  motor  are  the 
following:  the  belt  must  necessarily  be  very  long- 
often  nearly  or  quite  500  feet ;  it  must  be  very  strong, 
and  consequently  wide  and  heavy,  since  it  must  trans- 
mit the  entire  power  of  the  mill ;  the  expense  is  there- 
fore great,  and  the  tendency  to  stretch  greater  than  in 
a  short  belt ;  because  of  the  weight  and  length  the  op- 
eration of  shortening  and  tightening  the  belt  is  much 
more  difficult  than  in  ordinary  cases;  since  the  belt 
cannot  be  easily  slipped  from  one  pulley  to  another, 
the  use  of  fast  and  loose  pulleys  for  engaging  and  dis- 
engaging the  shafts  is  extremely  difficult.  The  advan- 
tages are  simplicity,  supposed  long  wear  (we  however 
doubt  very  much  the  truth  of  this,  since  the  belt  is  con- 
stantly bent  in  both  directions  and  run  on  both  sides 
upon  the  various  pulleys),  the  fact  that  the  driving- 
pulley  need  be  no  wider  than  is  necessary  to  carry  the 
one  belt,  and  economy  of  pulleys,  the  number  of  which 
is  less  than  if  the  power  of  each  shaft  was  obtained  by 
means  of  a  second  pulley  from  its  nearest  neighbor. 

Fig.  119  represents  a  three-story  mill,  in  which  each 
principal  shaft  is  connected  by  its  own  belt  directly 
with  the  driving-pulley.  Disadvantages  :  the  shop  is 
so  cut  up  by  the  many  belts  that  valuable  space  is  sacri- 
ficed ;  the  driving-pulley  must  be  wide  enough  upon 
its  face  to  carry  all  the  belts — in  the  figure  there  are 
seven  belts  ;  if  they  average  six  inches  wide  and  we  al- 
low one  quarter  of  an  inch  between  each  two  belts,  the 
face  of  the  driver  must  be  over  three  and  one  half  feet 
wide ;  the  use  of  fast  and  loose  pulleys  for  the  shafts 


APPENDIX. 


263 


is  rendered  difficult.  Advantages  :  each  belt  transmits 
the  force  of  one  shaft  only,  and  the  belts  may  therefore 
be  light ;  if  any  one  belt  breaks  it  may  be  removed  and 


FIG.  119. 


the  remaining  shafts  driven  as  if  no  accident  had  oc- 
curred ;  each  belt  may  be  made  large  or  small,  accord- 
ing as  it  has  heavy  or  light  work  to  perform. 


264 


BELTS  AND  PULLEYS. 


In  Fig.  1 20,  we  show  a  section  of  a  three-story  mill 
driven  in  the  manner  most  common  at  the  present  time 
throughout  this  country.  A  main  driving-belt,  heavy 


FIG.  120. 


enough  to  transmit  the  entire  work  of  the  mill,  runs 
from  the  motive-pulley  A  to  the  nearest  shop-shaft  B. 
From  the  latter  shaft  to  the  third-story  main  shaft  runs 
a  belt  sufficiently  strong  to  transmit  the  work  of  the 


APPENDIX.  265 

third  story.  The  other  shafts  on  each  story  are  con- 
nected by  separate  belts  each  with  its  nearest  neigh- 
bor, as  shown  in  the  figure.  In  this  arrangement  the 
belts  are  all  overhead  and  out  of  the  way,  except  two 
which  run  close  to  the  ends  of  the  building.  Thus  no 
valuable  space  is  used  up  by  the  belts.  Fast,  and  loose 
pulleys  may  easily  be  used,  because  none  of  the  belts 
(except  the  driver)  pass  over  the  driving-pulley.  This 
mode  of  transmitting  power  is  open  to  the  objection 
that  the  breakage  of  one  of  the  principal  belts  causes 
a  stoppage  of  several  shafts, — for  instance  if  the  hori- 
zontal belt  from  the  pulley  B  breaks,  the  entire  second 
story  is  thrown  out  of  gear ;  but  its  other  advantages 
more  than  compensate  for  this  risk,  and  it  has  there- 
fore come  to  be  the  favorite  in  most  of  our  shops  and 
factories. 


INDEX. 


Absence  of  early  records i 

Age  of  the  belt  and  pulley 3 

"  American  Machinist"   60 

Angle  between  middle  planes 31 

—  between  shafts 31 

Anti-friction  bearing 260 

Arc  embraced  by  belt 83 

Arms,  curved 167 

— ,  examples  of 168 

— ,  formulas  for 167 

— ,  method  of  drawing 169 

— ,  of  cable- pulleys 226 

— ,  profiles  of 169 

— ,  of  pulleys 166 

— ,  straight 167 

Arnold's  rule v 

"  Arts  and  Sciences  of  the  Ancients"  2 

Axes 28 

B 

Babylon 9 

Base  of  Naperian  Logarithms 84 

Belt  buckle 74 

—  hooks 71 

Belts,  canvas 66 

— ,  Clissold's 193 

— ,  crossed 7 

— ,  double 66 

— ,  flax 66 

— ,  gut  66 

— ,  half-crossed 31 

— ,  hemp 66 

— ,  leather 65 

— ,  metallic 192 


Belts,  open 1 1 

—  over  covered  pulleys 116 

— ,  rawhide 66 

— ,  rubber 259 

— ,  sheet-iron 67 

— ,  vulcanized-rubber 66 

—  without  guides  29 

—  with  pulley-guides 32 

Bending  strain  171 

Binomial  formula 80 

Briggs  and  Town e 248 

Breaking  strength  of  leather 92 

—  of  rawhide 66 


—  of  rope-belts 188 

—  of  vulcanized  rubber 141 

Broken  joints 66 


Cables,  deflections  of 207 

— ,  diameter  of 200 

— ,  examples  of 204 

— ,  formulas  for 200 

— ,  metallic 196 

— ,  number  of  strands  of 197 

— ,  number  of  wires  of 197 

— ,  rules  for 197 

— ,  strength  of 200 

— ,  tables  for 201-203 

— ,  tensions  of 196 

Cast-iron  shafts 176 

Circumference n 

— ,  examples  of 25 

— ,  formulas  for n 

— ,  rules  for n 

Circumferential  velocity 13 


268 


INDEX. 


PAGE 

Cleat-fastening 74 

Clissold's  belt ,      ....  193 

Coefficient  of  friction  of  cables 199 

—  of  jointed  chain-belts *93 

—  of  leather  over  cast-iron 86 

—  of  leather  over  leather 116 

—  of  rope-belts 187 

—  of  rubber  over  cast-iron 142 

—  of  rubber  over  leather 156 

—  of  rubber  over  rubber 156 

Coinciding  axes  28 

Common  logarithms . .  85 

Comparison  of  formulas vii 

—  of  leather  and  rubber 67 

Conditions  necessary  for  maintain- 
ing belt  on  pulleys 28 

Continuous  motion 7 

—  speed  cones 62 

Cooper,  J.  W 71 

Cores  of  metallic  cables 198 

Cork 225 

Covered  pulleys "6 

Crossed  axes 28 

Crossed  belt 7 

Curved  arms 167 


Decreasing  pulley-train 22 

Deflections  in  cables 207 

Device  for  changing  motion 254 

—  for  increasing  speed 257 

—  for  increasing  tension 255 

—  for  obtaining  intermittent  motion  258 

—  for  obtaining  opposite  motions..  255 

—  for  obtaining  variable  motion.   .  259 

—  for  putting  on  cables 241 

Diameter,  examples  of 25 

— -,  formulas  for  12 

—  of  cables .  202 

—  of  shafts 176 

Difficulties  found  in  belting 76 

Dimensions  of  pulley-supports 234 

Direction  of  rotation  n 

Distance  between  pulleys 31 

—  between  pulley-supports 231 

Double  belts    66 

—  curved  arms 169 


Double  lacing 71 

Dynamometer 77 


E 

Elasticity  of  cables 198 

Engaging  and  disengaging 180 

Entire  simplicity  impossible viii 

Enbank,  Thomas ..       i 

Examples  of  arms 168 

—  of  circumference 25 

Examples  of  continuous  speed  cones  64 

—  of  deflection 209 

—  of  diameter 25 

—  of  diameter  of  cables 204 

—  of  diameter  of  shafts 172 

—  of  greatest  tensions 90 

—  of  horse-power 27 

—  of  inclined  transmission 219 

—  of  increased  tension 214 

—  of  jointed  chain-belt 194 

—  of  keys 163 

—  of  length  of  belts 49 

—  of  nave 161 

—  of  power 27 

—  of  pulley-train 26 

—  of  radius 25 

—  of  revolutions . .     25 

—  of  rim 160 

—  of  rope-belts 189 

—  of  speed-cones  55 

—  of  transmission  with  pulleys  near 
together 222 

Examples  of  velocities 26 

—  of  weight  of  pulleys 165 

—  of  weight  of  principal  pulleys. . .  236 

—  of  width  of  leather  belts  over  cast- 
iron  pulleys 93 

Examples  of  width  of  leather  belts 
over  leather-covered  pulleys 118 

Examples  of  width  of  rubber  belts 
over  cast-iron  pulleys 154 

Examples  of  width  of  rubber  belts 
over  leather-covered  pulleys 157 

Expense  of  belting Hi 

Experiments  with  leather  over  iron  243 

—  with  leather  over  leather 249 

—  with  rubber  over  iron 251 


INDEX. 


269 


PAGE 

Experiments     with     rubber     over 

leather 252 

Extracts  from  letters iv 


Fast  and  loose  pulleys .   .  179 

Fastenings 68 

Fireless  period  4 

Fire-machine 4 

First  human  necessity 4 

—  machine 4 

—  transformation 5 

Fixing-keys 162 

Flax  belts 66 

Foot-pound  23 

Formulas  for  arms 167 

—  for  arms  of  cable-pulleys 226 

—  for  cable  diameters 200 

—  for  circumference n 

—  for  deflections 207 

—  for  diameter 12 

—  for  distance  between  pulleys. . . .     31 

—  for  face-width 159 

—  for  fixing-keys 163 

—  for  horse-power 24 

~~  for  inclined  transmission 219 

—  for  increased  tension 213 

—  for  jointed  chain-belts 193 

—  for  length  of  belts  47 

—  for  nave 161 

—  for  power — 20 

—  for  pressure  on  axes 236 

—  for  pulley -supports 234 

—  for  radius 12 

—  for  ratio  of  powers 22 

—  for  ratio  of  revolutions 14 

—  for  ratio  of  velocities 18 

—  for  revolutions , , 14 

—  for  rim 160 

—  for  rope-belts 187 

—  for  shafts. . 172 

—  for  speed-cones 54 

—  for  tensions 8.4 

—  for  tensions  in  cables 199 

—  for  velocities 15 

—  for  weight  of  pulleys  165 

—  for  weight  of  principal  pulleys. . .  236 


PAGE 
Formulas    for    width    of    leather 

belts 93-"4 

—  for  width  of  rubber  belts 150 


Godin's  belt. 192 

Graphical  method ...  60 

Greatest  tension 88 

Gum 116 

Gut  belts 66 

Gutta  percha . .  224 

H 

Half-crossed  belts 31 

Haswell's  rule vii 

Height  of  cable  above  ground 212 

Hemp  belts 66 

Herland's  tool 241 

Hirn  brothers  196 

Holes  for  lacing 68 

Horizontal  transmissions 207 

Horse-power 23 

"  Hydraulics  and  Mechanics" i 

Hyperbolic  logarithms 23 

I 

Inclined  transmissions    217 

Increased  tension 212 

Increasing  pulley-train 22 

Inferior  limit  of  separation  of  pul- 
leys   197 

Integral  calculus  84 

Intermediate  pulleys 230 

—  stations 232 

Intersecting  axes 28 

Intestines  for  belts 66 


Jointed  chain-belts 192 

"  Journal  of  Franklin  Institute"  ...  246 


Kenedy's  translation 4 

Keys 162 


Kilogram 200 

"  Kinematics  of  Machinery" . , 4 


2/0 


INDEX. 


Lacing 68 

Lack  of  knowledge  of  belting iv 

—  of  space 43 

Leather  belts 65 

— ,  examples  of 93 

— ,  formulas  for 93-IJ4 

— ,  tables  of 88-138 

Leather-covered  pulleys 115 

Logarithms,  common 85 

— ,  Naperian 84 

Long  belts 262 

M 

Material  of  belting 65 

Median  line 28 

Metallic  belts 192 

. —  cables  196 

Middle  plane 28 

Methods  of  arranging  pulleys  260 

Method  of  tracing  cable-curves  ...  221 

Middle  plane 28 

Mill-shafts 260 

Mutilated  pulley 258 

N 

Naperian  logarithms 84 

Nave  of  cable-pulleys 226 

—  of  pulleys 161 

Nineveh 8 

Nystrom's  formula vi 

O 

Open  belt 7 

Origin  of  belt  and  pulley 3 

Oscillating  motion 5 

P 

Parallel  axes 28 

Pedestals.  239 

Permissible  deviation  44 

Power,  examples  of 27 

— ,  formulas  for 20 

— ,  ratio  of 22 

Pressure  on  axes 235 

Primitive  lathe,  drill,  etc 5 

—  water-wheel 8 

Probable  origin  of  pulley 5 

Profiles  of  arms     169 


PAGE 

Proper  disposition  of  pulleys  28 

Pulley  arms 166 

— ,  cable . .  226 


-,  flanged 160 

-  nave 161 


--rim -. 159 

— ,  rounding  of 159 

— ,  split 163 

—  supports 230 

—  train 21 

—  with  light  arms 238 

R 

Radius,  examples  of 25 

— ,  formulas  for 12 

Ratio  of  circumferences 12 

—  of  power  22 

—  of  revolutions 14 

—  of  velocities 19 

Rawhide  belts 66 

Reserve  cables 228 

Resistance  to  slipping 73 

Reuleaux,  Prof vi,  28 

Reversing 182 

Revolutions , 14 

Rim  of  cable-pulleys 203 

—  of  pulleys 159 

Robertson 2 

Rollin 2 

Rope-belts 185 

Rosin 116 

Rotation it 

Rouiller's  belt. 192 

Rounded  fillies 28 

Rules  for  arms    167 

—  for  belts  with  pulley-gufdes 32 

—  for  circumference. .      n 

—  for  diameter i\ 

—  for  distance  between  pulleys —  31 

—  for  horse-power — 24 

—  for  power 20 

—  for  proper  disposition  of  pulleys.  28 

—  for  radius I2 

—  for  ratio  of  circumferences 12 

—  for  ratio  of  powers 22 

—  for  ratio  of  velocities 19 

—  for  revolutions J4 

—  for  shaft-diameters 172 


INDEX. 


2/1 


Safe  shearing  stress 142 

—  working  stress,  leather 92 

—  working  stress,  rubber 141 

Scale  for  cable-curves 220 

Schaff house 228 

Shafts 171 

—  of  mines 233 

Sheet-iron  pulleys 238 

Shop-shafts 260 

Shortening 68 

Single  lacing 69 

Size  of  pulleys 24 

Slipping 45 

Slow  growth  of  belting 6 

Smith,  C.  A 60 

Speed-cones... 51 

Spinning-mills 44 

Split  pulleys 163 

Stations 232 

Station  pillars 238 

Steel  cables 203 

— -  shafts 176 

Stepped  cones 65 

Strength  of  gut 66 

—  of  leather 75 

—  of  rawhide 66 

—  of  vulcanized  rubber 141 

Swedish  iron 203 


Table,  deflections 210 

— ,  dimensions  of  pulley-supports..  234 

— ,  formulas  for  leather  belts 96-129 

— ,  formulas  for  rubber  belts. . .  142-149 

— ,  greatest  tensions 90,  118,  157 

— ,  increased  tension 214 


Table,  metallic  cables 210 

— ,  number  of  arms 167 

— ,  shaft-diameters 176 

— ,  tensions  for  leather 88,  117 

— .  tensions  for  metallic  cables 201 

— ,  tensions  for  rope-belts 188 

— ,  weight  of  pulleys 165 

— ,  widths  of  leather  belts no,  135 

— ,  widths  of  rubber  belts 151 

Tensions  in  cable-wires 200 

—  in  belts 79 

—  in  inclined  transmissions 220 

Thickness  of  rubber  belts 140 

Tightening-  pulley 179 

Torsional  strain 171 

Tower  of  Babel 9 

Transmission  by  cable  with  pulleys 

near  together 222 

Transmission  with  inclined  cable...  217 
"  Treatise  on  Toothed  Gearing". .     169 

U 

Uncertain  origin  of  belt  and  pulley      3 
Unwin's  formula vi 


Velocities 13 

Vulcanized-rubber  belts 66 

W 

Weakest  part  of  belt 92 

Weight  of  pulleys 164 

Wrought-iron  shafts , .  176 


Zeigler's    machine    for  putting  on 
metallic  cables 241 


JEWELL  BELTING  CO. 

(Successors  to  P.  Jewell  &  Sons), 


MANUFACTURERS    OF 


Leather  Belting 


AND 


LACE    LEATHER, 

Hartford,    -      -      -    Conn, 


SHULTZ  BELTING  COMPANY, 

Manufacturers  of  Shultz  Patent 

Fulled  Leather  Belting,  Lace  and  Picker  Leather, 


OUR  BELTING  is  made  of  Leather,   tanned  on  the  surface  only;  the  interior 
(which  is  the  fibre  and  strength  of  the  hide)  is  not  tanned,  but  Eawhide  fulled 
and  softened  by  our  patent  process.     Our  belting  is  more  pliable,  and  hugs  the 
pulley  better  and  transmits  more  power  than  any  other  Belt     It  does  net  pull  out 
ar  the  laceholes  or  rivets.    It  stretches  less  thun'any  other  Belt.    Jt  works  equally 
well  for  the  largest  "Driving  Belts  or  for  the  fastest  running  machinery  and  smallest 
'  pulleys.    Our  LACE  LEATHER  is  made  of  Rawhide,  by  our  patent  process,  with- 
out any  tanning,  and  is  stronger  and  will  wear  better  than  any  other.    We  also 
make  the  best  Picker  Leather  and  Belt  Grease  in  the  country. 

Cor.  Bismarck  and  Barton  Streets,  St.  Louis,  Mo. 


FRANK  PEIRCE,  128  Pearl  Street, 

Boston,  Mass. 
JAMES  GARNETT,  140 North  Third 

Street,  Philadelphia,  Pa. 
U.  BAIRD   MACHINERY   CO.,   75 

Water  Street,  Pittsburg,  Pa 
CURTIS  &  CO  MFG.  CO.,  40  Franklin 

Street,  Chicago,  111. 
J.  L.  LINDSAY,  Richmond,  Va. 
PARKIN  &  BOSWORTH,  Cleveland, 

Ohio. 
SHAW,  KENDALL  &  CO.,  Toledo,  0. 


ROGERS,  WILLIS  &  CO.,  St.  Paul, 
Minn. 

JANNEY,  SEMPLE  &  CO.,  Minne- 
apolis, Minn. 

A.  G.  AUSTIN  &  CO.,  Terre  Haute, 
Ind. 

GEN.  FRED.  N.  OGDEN,  New  Or- 
leans, La. 

J.  H.  COFFIN  &  CO.,  Memphis,  Tenn. 

R.  L.  WATKINS  &  SONS,  Chatta- 
nooga, Tenn. 

A.  D.  TRUESDELL,  Warren,  Ohio. 


STAMPED  BELOW 


RENEWED  BOOKS  ARE  SUBJECT  TO  IMMEDIATE 
RECALL 


OCT  *  6 1964 
RET,  OC 

OCT  2  6 1966 
26  NOV  B6 


UCD  LIBRARY 

DUE  JAN  2  6  1970 
JAN  1 4  REC'D 


LIBRARY,  UNIVERSITY  OF  CALIFORNIA,  DAVIS 

Book  Slip-50m-8,'63(D9954s4)458 


Cromwell,   J.H. 
Treatise  on  belts  and 


TJ1100 


TJilOO 


294424 


